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Universitat de les Illes Balears Departament de Física

UIB

PhD thesis

Programa de doctorat dels Estudis Oficials de Postgrau:

Doctorat de Fısica

Gravitational waveobservation of compact binaries

Detection, parameter estimation and template accuracy

Miquel Trias Cornellana

(Advisor: Dr. Alicia M. Sintes)

November 2010

UIB

Universitat de les Illes Balears

i

Gravitational wave observation of compact binaries:detection, parameter estimation and template accuracy

byMiquel Trias Cornellana

THESISPresented to the Physics Department of

Universitat de les Illes Balearsin Partial Fulfillmentof the Requirements

for the Degree of

DOCTORin

PHYSICS

November 2010

ii

Gravitational wave observation of compact binaries: detection, parameter es-timation and template accuracy

Miquel Trias ([email protected])Departament de Fısica, Universitat de les Illes Balears,Carretera de Valldemossa km. 7.5, 07122 Palma de Mallorca, Spain

PhD Thesis

Supervisor: Dr. Alicia M. Sintes

2010, Miquel Trias CornellanaUniversitat de les Illes BalearsPalma de Mallorca

This document was typeset with LATEX 2ε

iii

La directora de tesi Alicia M. Sintes Olives, Professora titulard’universitat de la Universitat de les Illes Balears, adscrita alDepartament de Fısica, certifica que aquesta tesi doctoral haestat realitzada pel Sr. Miquel Trias Cornellana, i perque enquedi constancia escrita, signa la present

a Palma de Mallorca, 29 de Novembre de 2010,

Dr. Alicia M. Sintes Miquel Trias

Resum (en catala)

En aquesta tesi es presenten els resultats obtinguts en tres lınies de recerca diferents,totes elles relacionades amb l’analisi de dades per a la deteccio directa d’ones gravitatoriesemeses per sistemes binaris d’objectes compactes (forats negres, estels de neutrons, nanesblanques) de massa similar.

Per una banda, hem estudiat quina sera l’estimacio de parametres que es podra fer quans’observin les ones gravitatories emeses pels xocs entre dos forats negres supermassius(normalment, situats als centres de les galaxies) en la seva fase inspiral amb el futur detec-tor interferometric espacial d’ones gravitatories, LISA. En particular, estudiem l’impacteque te la inclusio de tots els harmonics de la senyal en l’estimacio de parametres, i hocomparem amb el resultat classic alla on nomes es considerava l’harmonic dominant dela senyal (el corresponent a una frequencia 2forb); veure Cap. 4. Els resultats obtingutsconfirmen la gran importancia d’emprar la senyal completa (amb tots els harmonics),basicament per dos motius: en primer lloc, ja que incrementen el rang de masses en elque LISA podra detectar senyals d’ones gravitatories procedents d’aquests objectes; peroprincipalment, perque la seva inclusio augmenta la riquesa dels detalls de la senyal re-buda, de manera que es fa mes facil distingir entre dues senyals amb distints parametresi per tant, es redueixen significativament (fins a diversos ordres de magnitud) els errorsen la seva estimacio. A consequencia d’aquest important resultat, tambe hem estudiat elnombre esperat de fonts que LISA detectara cada any amb un cert error donat [tant en ladeterminacio de la distancia, com de la posicio al cel], i en funcio del model de formaciode galaxies que es considera per al nostre Univers; i per altra banda, la precisio amb laque podrıem mesurar l’equacio d’estat de l’energia fosca a partir d’una unica observacio deLISA (veure Cap. 5). Les conclusions d’aquests dos darrers estudis es que LISA observaracada any, unes 20 fonts (nomes de xocs entre forats negres supermassius) amb precisio defins al 10% en la mesura de la distancia, 10 fonts amb una resolucio al cel millor de 10 deg2

i el que es mes interessant, esperem observar 1 − 3 fonts amb una exel·lent precisio, tanten la determinacio de la posicio al cel (millor que 1 deg2) com de la distancia (millor del’1%); tot plegat ens permetra emprar LISA per a realitzar cosmografia de precisio, semprei quan puguem eliminar l’efecte de lents gravitatories creat pels objectes que hi ha entrenosaltres i les fonts que volem estudiar a 0.5 < z < 1.

Per altra banda, hem desenvolupat un algorisme de cerca de senyals gravitatories proce-dents de sistemes binaris estel·lars situats dins la nostra propia galaxia. Aquest algorismeesta basat en la interpretacio Bayesiana de la probabilitat i empra tecniques d’integracio

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per Monte Carlo mitjancant cadenes de Markov (MCMC) de manera que ens permet, nonomes detectar les senyals, sino que alhora estimem la distributicio de probabilitat delsparametres que caracteritzen la font. S’espera que el detector LISA observi simultaniamentdesenes de milers d’aquestes senyals, aixı doncs, la distincio entre cadascuna d’elles es unatasca que es realitza amb l’analisi de dades posterior. En aquesta tesi, hem implementatun metode que serveix tant per cercar una unica senyal dins les dades, com un nombrefixat d’elles (veure Cap. 6). A mes, hem desenvolupat un algorisme totalment general i quepreserva el caracter Markovia de la cadena que es genera, basat en el Delayed Rejectionper tal de mostrejar eficientment funcions que presenten una estructura multimodal (es adir, que tenen diversos maxims relatius separats per una certa distancia); veure Cap. 7.Aquest tipus d’estructures son molt comuns en un nombre molt divers d’aplicacions delMCMC (no nomes en l’area d’analisi de dades d’ones gravitatories, ni tan sols unicamenten l’area de Fısica) i la presencia de diversos maxims de la funcio pot reduir consider-ablement l’eficiencia dels metodes de mostreig. L’algorisme que nosaltres hem dissenyat,esperem que pugui solventar aquest tipus de situacions, que son particularment comuns irellevants en l’analisi de dades per LISA.

Finalment, la tercera lınia de recerca portada a terme durant el doctorat ha consistiten estudiar el rang de validesa dels models de patrons d’ones gravitatories (emeses persistemes binaris compactes) mes rapids de generar computacionalment que existeixen,pero que alhora contenen certes aproximacions. El context d’aquest estudi es troba en lescerques que actualment es fan amb els detectors interferometrics terrestres (LIGO i Virgo)d’una de les fonts mes prometedores de ser detectada: el xoc entre dos forats negres (oestels de neutrons) de fins a 500M. Tots els metodes de cerca emprats es basen compararles dades mesurades, amb els patrons teorics que esperem que estiguin continguts dinsaquestes en cas de que hi hagi una senyal (matched filtering); com mes s’ajustin aquestspatrons a la senyal real, mes possibilitats hi ha de reconeixer-la d’entre el renou. Aixıdoncs, per una banda necessitem que els patrons emprats en la cerca siguin el mes precisospossible, pero per l’altra, resulta que hem cobrir tot l’espai de parametres, el que implicala generacio de molts d’aquests patrons, aixı que tambe requerirem que siguin rapids degenerar. En el Cap. 8 definim matematicament quins son els requisits mınims de precisioque han de satisfer els models de patrons d’ones gravitatories i estudiem per quin rangde masses es poden emprar els models mes rapids, aquells que directament ens donen unaexpressio analıtica tancada per als patrons. Les conclusions que obtenim son que aquestsmodels rapids ens garanteixen (per sistemes amb una relacio de masses entre 1:1 i 4:1) noperdre mes de 6% de les possibles senyals quan es consideren els detectors actuals i no mesdel 9% quan es consideren detectors avancats. Per sistemes amb un quocient de massesmajor i en qualsevol cas en que un estigui interessat no nomes en detectar la senyal, sinotambe en extreure informacio creıble sobre els parametres fısics mesurats; aleshores seranecessari emprar models mes precisos.

Acknowledgements

I would like to express my sincere gratitude to my thesis advisor, Dr. Alicia M. Sintes, forintroducing me to research and for her guidance and help during these four years.

I would like to especially thank Prof. Alberto Vecchio for all the invaluable lessons, advicesand ideas that I received from him, and also for his hospitality during the many times thatI visited the University of Birmingham. He truly has been like a co-advisor to me duringall these years.

I also would like to thank Prof. Thibault Damour for many useful discussions and sugges-tions, and for his hospitality during my visit at the Institut des Hautes Etudes Scientifiques(IHES) in the autumn 2009. From him I have learned the power of thoroughness and sim-plicity in science.

I am grateful to Dr. John Veitch for “converting” me to Bayesian statistics and showingto me how powerful it can be, and also for his infinite patience in the development andimplementation of the algorithms described in Chapters 6 and 7. I thank Dr. AlessandroNagar for all the useful discussions we had about the physics behind a compact binarycoalescence, also for his (and his wife’s) hospitality during my visit at the IHES andspecially for his encouragement and support during this last year.

I would like to thank all my colleagues from the UIB Agencia EFE and the Relativity andGravitation group at the UIB for all their support and specially to Dr. Sascha Husa formany valuable discussions and suggestions.

I thank Dr. Alexander Stroeer for sharing his MCMC code, which facilitated the develop-ment of the software used in Chapter 6. I am grateful to Dr. Badri Krishnan, Dr. StanislavBabak, Dr. Edward K. Porter and Dr. Jonathan R. Gair for all the discussions we had aboutextreme mass ratio inspiral signals while I was visiting the Albert Einstein Institute (AEI)in the spring 2007. I also would like to thank Prof. Bangalore S. Sathyaprakash, Dr. ChrisVan Den Broeck and all the members of the LISA Performance Evaluation Taskforce1 formany useful discussions about LISA parameter estimation and higher harmonics.

I thank the organizers of the LIGO Astrowatch program and all the scientists, operatorsand other ‘astrowatchers’ of the LIGO Hanford Observatory that had the patience to

1http://www.tapir.caltech.edu/dokuwiki/lisape:home

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teach me how to operate an interferometer and more importantly that became friends inthe middle of that desert.

I would like to thank all the institutions where I spent some of the time of my doctor-ate for hospitality and the possibility of meeting new people: the University of Birming-ham (Prof. Alberto Vecchio), the AEI (Prof. Bernard F. Schutz, Dr. Badri Krishnan andDr. Maria A. Papa), the LIGO Hanford Observatory (Prof. Fred Raab and Dr. MichaelLandry) and the IHES (Prof. Thibault Damour). In all of them I took many valuablelessons.

Volia agrair el suport i consells rebuts per part de tota la meva famılia: pares, germanes,padrins, . . . Heu estat sempre alla en els moments difıcils, preocupant-vos i assessorant-meper trobar el camı que m’ha permes arribar fins aquı.

Dono les gracies a tots els meus companys de voleibol de la UIB per la seva amistat iper proporcionar-me aquestes dues hores setmanals d’adrenalina i desconnexio que tanthe agraıt, sobretot en aquests darrers mesos d’escriptura. En especial, hem d’estar agraıtsa Ruben Santamarta per iniciar i mantenir viva aquesta activitat.

Tampoc m’oblido dels meus amics. Aquesta es la primera de cinc tesis que comencarencom a discussions de cafe i gelat ara ja fa gairebe deu anys per alla per Porto Pı; graciesToni, Vıctor, Pau i Javi per tot el suport, forca i afecte que m’heu donat durant aquestsanys, sabeu que es recıproc! Moltes gracies tambe a Aina, Esther i Sergio, parte de estatesis tambien es gracias a vuestro apoyo.

Finalment, el meu agraıment mes especial es per na Llucia, la meva millor amiga, companyai al·lota des de fa deu anys i mig. Amb ella he compartit tots els instants dels darrers quatreanys; hem viscut plegats els millors moments del doctorat [el meu primer article, creuantmitja Europa en cotxo, vivint tres mesos enmig d’un desert america, el dia en que elDR a la fi va funcionar, els shifts al detector de Hanford, el congres a Nova York. . . ], perosobretot sempre hi ha estat quan mes ho he necessitat [el primer cap de setmana en aquellacasa de Birmingham, els deadlines dels MLDCs, els dies previs a la xerrada als Amaldi,l’escriptura de la tesi. . . ]. Al mirar aquesta tesi, veig un recull de les experiencies que hemviscut plegats.

This thesis was done within the Doctorate Program of the Physics Department of the Universitat

de les Illes Balears with the financial support of a predoctoral grant (FPI06-43114641Z) given

by the Govern de les Illes Balears, Conselleria d’Economia, Hisenda i Innovacio during the first

two years; and a FPU grant (AP2007-01365) given by the Spanish Ministry of Science during the

last two years. The work presented here was also partially supported by the Spanish Ministry of

Science research Projects No. FPA-2004-03666, FPA-2007-60220, HA-2007-0042, CSD-2007-00042,

and CSD-2009-00064; the European Union FEDER funds; and the Govern de les Illes Balears,

Conselleria d’Economia, Hisenda i Innovacio.

Contents

Preface xiii

I Introductory notions 1

1 Introduction 3

1.1 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Einstein’s equations for a weak gravitational field . . . . . . . . . . . . . . . 5

1.1.2 Independent components of hµν . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.3 Effect of gravitational waves on free particles . . . . . . . . . . . . . . . . . 7

1.1.4 Production of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Astronomical gravitational wave sources . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Direct detection of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Cryogenic resonant-mass detectors . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.2 Interferometric gravitational wave detectors . . . . . . . . . . . . . . . . . . 14

1.4 Compact Binary Coalescences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Observed GW signals: from emission to detected signals 23

2.1 Emitted gravitational wave signals from compact binaries . . . . . . . . . . . . . . 24

2.1.1 Spin (s = −2) weighted spherical harmonics . . . . . . . . . . . . . . . . . . 25

2.1.2 General expression for h+ and h× . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Detected gravitational wave strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.1 Gravitational wave strain in the detector’s coordinates frame . . . . . . . . 36

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x Table of contents

2.2.2 Apparent sky location and polarization from detector’s motion . . . . . . . 39

2.2.3 Doppler shift due to relativistic effects and relative motion of the source . . 40

2.3 Measured GW strain as a sum of cosine functions . . . . . . . . . . . . . . . . . . . 42

2.4 Residual gauge-freedom in the measured GW strain . . . . . . . . . . . . . . . . . 43

3 GW data analysis for CBCs 49

3.1 Gravitational wave signals buried in noise . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Detection and parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 Frequentist framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.2 Bayesian framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.3 Fisher information matrix formalism . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Modeling gravitational waveforms from Compact Binary Coalescences . . . . . . . 59

3.3.1 Stationary Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4 Visualizing gravitational waveforms from CBCs . . . . . . . . . . . . . . . . . . . . 66

3.5 Effective/characteristic amplitudes and observable sources . . . . . . . . . . . . . . 70

3.5.1 Analytical calculations within the Newtonian approximation . . . . . . . . 73

3.5.2 Plotting the effective/characteristic amplitudes . . . . . . . . . . . . . . . . 75

II Original scientific results 85

4 Impact of higher harmonics on parameter estimation of SMBHs inspiral signals 87

LISA observations of SMBHs: PE using full PN inspiral wvfs. [PRD, 77 024030 (2008)] 87

LISA parameter estimation of SMBHs [CQG, 25 184032 (2008)] . . . . . . . . . . . . . 87

Massive BBH inspirals: results from the LISA PE taskforce [CQG, 26 094027 (2009)] . . 87

5 Measuring the dark energy equation of state with LISA 133

Weak lensing effects measuring the dark energy EOS . . . [PRD, 81 124031 (2010)] . . . . 133

6 Searching for galactic binary systems with LISA using MCMC 151

MCMC searches for galactic binaries in MLDC 1B data sets [CQG, 25 184028 (2008)] . 151

Studying stellar binary systems with LISA using DR-MCMC [CQG, 26 204024 (2009)] . 151

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7 Delayed Rejection Markov chain Monte Carlo 175

DR schemes for efficient MCMC sampling of multimodal distrib. [arXiv: 0904.2207] . 175

Addendum to Chapter 7. Toy examples to illustrate the efficiency of DR MCMC chains 209

8 Studying the accuracy and effectualness of closed-form waveform models 219

Accuracy and effectualness of waveforms for nonspinning BBHs [PRD, 83 024006 (2011)] 219

9 Conclusions 251

List of Acronyms 257

Preface

This thesis is divided in two different parts, clearly separating what corresponds to Intro-ductory notions from the Original (published) scientific results. Since most of the resultsproduced during the doctorate were already published in refereed international scientificjournals, or about to be accepted, we decided that this second part would consist in theinclusion of the publications produced by the applicant. There are several reasons behindthis decision that, in our opinion, benefit all the parts that will ‘interact’ with this PhDthesis. On one hand, it gives more time to the applicant to focus on the introductory no-tions and on better understanding the basics related with the field they have been workingon during the last years. On the other hand, it also easies the referee process as the origi-nal scientific results are written as they were originally published, instead of repeating thesame information with other words. And finally, it is also useful for future readers of thisthesis, as they can clearly distinguish between what are the side results only produced forthe thesis and what has been published in a refereed journal and therefore spread to thewide scientific community.

In the first three chapters of the thesis, comprising Part I, we first give (Chapter 1) abrief introduction to the research field and then, in Chapters 2 and 3 we try to deriveand understand the basic expressions that form the well-stablished basics of GW dataanalysis, at the same time that justify the scientific studies that are presented in Part II.Of course, all the results derived in Part I can be found in many other publications, andfor this reason they would never be accepted in a refereed journal; however, we considerthat the knowledge acquired from deriving all the expressions from scratch and using auniform notation can be extremely useful for the applicant in his future scientific careerand we wish that also this material could be useful for future students entering this field.

The scientific results presented in this thesis (Part II) can be divided into three differentresearch lines (see Tab. 9.1 for a summary), all of them related to data analysis stud-ies of gravitational waves emitted by compact binary objects. Our main conclusions aresummarized in Chapter 9.

• First, we perform parameter estimation studies (Chapter 4) of supermassive binaryblack hole inspirals observed with LISA (the future gravitational wave space antenna)using post-Newtonian waveforms that include all the harmonics and we compare theoutput with the classical results where only the dominant (` = 2,m = ±2) waspresent. Then, we also study what are the consequences of this improvement on the

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xiv Preface

science that LISA will be able to do, in particular, we study what is the expectednumber of SMBH sources that LISA will observe with a particular error and also,the precision on the estimation of the dark energy equation of state (Chapter 5)

• Second, we develop a search method for galactic binary signals with LISA based onBayesian probability and Markov chain Monte Carlo techniques and apply it to anumber of simulated data sets containing one or several overlapping signals (Chap-ter 6). On the way, we shall face the problem of efficiently sampling a multimodaldistribution, and the solution we propose in Chapter 7 is a completely Markovianand fully general algorithm, based on a technique called Delayed Rejection, that wesuccessfully implement for the search of galactic binaries with LISA. We belief thatthis general method can be applied to a number of problems in a variety of fields.

• Finally, we study the accuracy and effectualness of some model waveforms used insearches for compact binary coalescences with ground-based interferometric detec-tors. In particular, we study the validity range of the fastest (but also approximated)waveform models either when they are used just for detection, or also for measure-ment purposes (Chapter 8).

As it is usual within the scientific community working on GR problems, in most of thesituations we will use so-called geometrized units in which c ≡ 1

[1 s = c m ' 3× 108 m

]and G ≡ 1

[1 s =

c3

Gkg ' 4× 1035 kg

], so we shall be effectively measuring length and

mass in units of time. Working with astrophysical problems, some significant relation tohave in mind are the following,

106M = 4.93 s ; 1 AU = 499 s ; 1 Gpc = 1.03× 1017 s

We also would like to take this opportunity to define some very common quantities thatappear throughout all the thesis and that sometimes it is useful to have all of them definedin the same place.

• Total mass: M ≡ m1 +m2;

• Mass difference: δm ≡ m2 −m1;

• Reduced mass: µ ≡ m1 ·m2

m1 +m2;

• Symmetric mass ratio: ν ≡ µ

M=

m1 ·m2

(m1 +m2)2[sometimes designated by η];

• Chirp mass: M≡Mν3/5;

• Characteristic velocity of a CBC inspiral: v ≡ (πfM)1/3;

• Post-Newtonian expansion parameter: x = (πfM)2/3 = v2;

• Characteristic velocity at the (test-mass particle) last stable orbit: vlso = 1/√

6.

Part I

Introductory notions

1

Chapter1Introduction

According to Einstein’s theory of general relativity (GR), compact concentrations of energy[e.g. neutron stars (NSs) and black holes (BHs)] should wrap spacetime strongly, andwhenever such an energy concentration changes shape, it should create a dynamicallychanging spacetime warpage that propagates out through the Universe at the speed oflight. This propagating warpage is called a gravitational wave — a name, that arises fromGR’s description of gravity as a consequence of spacetime warpage.

Although gravitational waves (GWs) have not yet been detected directly, their first indirectevidence was found in the observed inspiral of the binary pulsar PSR 1913+16, discoveredby Hulse and Taylor [1]. Taylor (and colleagues) demonstrated [2, 3] that the observed NSsare spiraling together at just the rate predicted by GR’s theory of gravitational radiationreaction; the computed and observed inspiral rates agree to within the experimental accu-racy, which is better than one per cent. In 1993, the Nobel Prize in physics was awarded toHulse and Taylor for their discovery. This is a great triumph for Einstein’s theory, howeverit is not a firm proof that GR is correct in all aspects. Other relativistic theories of gravity(i.e. compatible with special relativity) also predict the existence of GWs; and some ofthem predict the same inspiral rate for PSR 1913+16 as GR, to within the experimentalaccuracy [4, 5].

The emission of detectable GWs is related to catastrophic, high-energetic events in theUniverse, such as supernovae explosions, compact binary coalescences (CBCs), rapidlyspinning NSs or even the Big Bang. Some of these events have already been observedthrough their electromagnetic emission; although if we compare the energy emitted asGWs to the electromagnetic counter-part, it turns out that the gravitational radiationluminosity is many order of magnitudes larger. The reason behind is because spacetime isa very “stiff” medium, i.e. large amounts of energy are carried by GWs of small amplitude.This fact can be easily seen, for instance, considering one among the most promising sourcesfor GW detectors [it is also the source considered in all the studies of this thesis], whichis the (similar mass) compact binary coalescences (CBCs), including from stellar-masscompact systems (NSs and BHs), up to supermassive black holes (SMBHs). Within theNewtonian approximation, the GW luminosity (or “flux”), Fgw, and amplitude, agw, can

3

4 Chapter 1: Introduction

be written as [6]

Fgw, n =32c5

5Gν2v10 and |agw, n| = 4

νM

DLv2 , (1.1)

whereG and c are the gravitational constant and speed of light, respectively1;M = m1+m2

is the total mass of the system, ν = m1m2M2 is the symmetric mass ratio, DL is the luminosity

distance to the source and v = (πfM)1/3 is a characteristic speed of the two orbitingobjects. Taking a typical equal-mass (ν = 1/4) BH-BH coalescence (M = 20M) withinthe Virgo supercluster (DL = 30 Mpc) at its last stable orbit, LSO (vlso = 6−1/2), weobtain a GW luminosity Fgw, n ' 2× 1048 W ' 5× 1021L, whereas the GW amplitude(which directly represents the relative length changes induced by the GW) is |agw, n| =

2 |∆L|L ' 5× 10−21. These are typical values for the expected GW sources and they explainwhy we have been able to indirectly measure the effect of gravitational radiation in termsof energy losses, but it is so hard to directly observe them. Notice from Eq. (1.1) that theGW luminosity of a CBC at the last stable orbit (LSO) is independent of the total mass,i.e. the power emitted by two stellar-mass BHs is the same as a 107M − 107M system;what is different is the amount of time that such luminosity is held (the time scales ofmassive systems are much longer).

Recent years have seen a shift in the technologies used in GW searches, as the first genera-tion of large GW interferometers has begun operation at, or near, their design sensitivities,taking up the baton from the bar detectors that pioneered the search for the first directdetection of GWs. In particular, an international network of ground-based multi-kilometerscale interferometers is currently operating and fundings are already approved to builtan advanced generation of the current interferometers, which should provide an order ofmagnitude improvement in their sensitivity.

The direct detection of GWs will represent a confirmation of the GR’s predictions, butmore importantly, it will open the exciting new field of GW astronomy which will provideanswers to a number of questions in various different areas [7]. In particular, we expect toget answers to questions for

• Fundamental Physics and General Relativity: What are the properties of GWs? IsGR the correct theory of gravity, and is it still valid under strong-gravity conditions?How does matter behave under extremes of density and pressure?

• Astronomy and Astrophysics: How abundant are stellar-mass BHs? What is the cen-tral engine behind gamma-ray bursts? Do intermediate mass BHs exist? Where andwhen do massive BHs form and how are the connected to the formation of galaxies?How massive can a NS be and how is their interior? What is the history of starformation rate in the Universe?

• Cosmology: What is the history of the accelerating expansion of the Universe? Werethere phase transitions in the early Universe?

1In most situations throughout this thesis, we shall use so-called geometrized units in which G = c = 1;however, here they are written explicitly in order to be able to compute quantities in SI units.

1.1. Gravitational waves 5

1.1 Gravitational waves

GWs are oscillations of the spacetime propagating away from the source that generatedthem as waves at the speed of light. There is an enormous difference [8] between GWs,and the electromagnetic (EM) waves on which our present knowledge of the Universe isbased, and indeed are these differences what make the information brought to us by GWsto be very different from (almost “orthogonal to”) that carried by EM waves, i.e. it isusually said that GW astronomy will be a completely new window to observe the Universethrough.

• EM waves are oscillations of the EM field that propagate through spacetime; whereasGWs are oscillations of the “fabric” of spacetime itself.

• GWs are produced by coherent, bulk motions of huge amounts of mass/energy andthe wavelengths are comparable to, or larger than, the emitting sources; kilometricground-based detectors will observe GW signals within the λ ∈ [10, 105] km range,whereas the 5 × 106 km long future space antenna will have the observable rangewithin λ ∈ [106, 109] km; which, in both cases, correspond to the typical size of thesources. On the other hand, EM waves are almost always incoherent superpositions ofemission from individual electrons, atoms or molecules, with much smaller emissionwavelengths.

• EM waves are easily absorbed, scattered, and dispersed by matter. Despite the hugeamount of energy carried by GWs, they almost do not interact with matter, thisis why it is so hard to detect them, but at the same time this also means that theinformation from the emitting source is preserved almost unaltered.

1.1.1 Einstein’s equations for a weak gravitational field

The existence of GWs was a prediction of theory of GR, and from it one can derive themain (general) properties that characterize them. As an introductory chapter, here wejust present a brief derivation of the main properties of GWs, which are covered in muchgreater depth in Refs. [8–12]; this section has been written mainly following Ref. [13].

We shall start by considering a GW as a ‘weak’ perturbation of a flat spacetime, whichcan be expressed mathematically as

gαβ = ηαβ + hαβ , (1.2)

where ηαβ = diag(−1, 1, 1, 1) is the Minkowski metric of Special Relativity and |hαβ| 1for all α and β. The coordinate system in which the metric components of a ‘nearly’ flatspacetime can be written as in Eq.(1.2) is not unique. Actually, once it has been identifieda coordinates system in which the metric components can be written in such way, it ispossible to find an infinite family of other coordinates systems that also have the metriccomponents written as Eq. (1.2); for instance, the Lorentz and gauge transformations aretwo changes of coordinates that preserve such general expression. Then, it is possible todemonstrate2 that in a free space (i.e. Tµν = 0) and making the appropriated coordinate

2Since this is an introductory chapter, we shall not go through all the details needed to finally obtainthe wave equation for hαβ . See Refs. [9, 10] for further details.


transformations, Einstein’s equations for the propagation of a metric perturbation can bewritten as hµν = 0, or in other words,(

− ∂2

∂t2+∇2

)hµν = 0 . (1.3)

This Equation (1.3) is nothing but a waves equation with a propagation velocity equal to1 ≡ c, i.e. it is found that the metric perturbations propagate at the speed of light throughfree space.

1.1.2 Independent components of hµν

The simplest solution to Eq. (1.3) is a superposition of plane waves, that can be writtenas

hµν = Re [Aµν exp(ikαxα)] , (1.4)

where the constant components Aµν and kα are known as the wave amplitude and wavevector respectively. These two quantities are not arbitrary; instead, they must satisfy someconditions that shall reduce the number of independent components from 16 to only 2:

• Aµν is symmetric, since hµν is symmetric; this immediately reduces the number ofindependent components from 16 to 10. Also, it is easy to show that the wave vector,kα, is a null vector.

• From the Lorentz gauge condition [which has been applied to obtain Eq. (1.3)], itcan also be seen that the wave amplitude components must be orthogonal to thewave vector: Aµαk

α = 0, reducing the number of independent components from 10to 6.

• It can be shown [9, 10] that the choice of coordinates that satisfy the Lorentz gaugeis not unique. This provides the freedom to choose a coordinates system in whichhµν , and therefore Aµν , can be written in an even more simplified way. In particular,we shall work with the transverse-traceless gauge3 (TT), where four more amplitudecomponents are fixed, leaving only 2 independent components.

Hence, after all these considerations and arbitrarily setting the propagation direction ofthe GW to be parallel to the z-axis, we have that

h(tt)µν = A(tt)µν cos [ω(t− z)] , (1.5)

where only two amplitude components are independent

A(tt)µν =

0 0 0 0

0 A(tt)xx A

(tt)xy 0

0 A(tt)xy −A(tt)

xx 00 0 0 0

. (1.6)

3With this choice of coordinates, one finds that the trace of Aµν is null, besides that their componentsare perpendicular to the wave’s propagation direction (i.e. it is a transverse wave).

1.1. Gravitational waves 7

€

Axx(TT ) ≠ 0 ; polarization

€

Axy(TT ) ≠ 0 ; polarization

Figure 1.1: Cartoon of the effects that the two independent polarizations of a GW traveling in thedirection perpendicular to the paper sheet produce over an annular distribution of mass. The top panel isobtained by setting A

(tt)xx 6= 0 ; A

(tt)xy = 0 and it corresponds to the ’+’ polarization; whereas in the bottom

panel, we are setting A(tt)xx = 0 ; A

(tt)xy 6= 0, getting the ’×’ polarization. [Diagrams: own production based

on [13]]

1.1.3 Effect of gravitational waves on free particles

We have just obtained a solution to Einstein’s equations for a metric perturbation propa-gating in free space. Now, we want to understand the effect that a GW produces when itpasses through matter and hence, to understand the significance of the two independent

amplitude components, A(tt)xx and A

(tt)xy , obtained in Eq. (1.5).

A free particle initially at rest, will remain at rest indefinitely. However, ‘being at rest’ inthis context simply means that the coordinates of the particle do not change when a GWpasses through. What will manifest the effect of the gravitational radiation is the changein the proper distance between two free particles. Hence, the effect of GWs is to modify theproper distance between two free masses, and indeed the relative length change is directlyrelated to the perturbation’s amplitude, ∆L

L = h2 . It is essentially this change in the proper

distance between test particles what GW detectors attempt to measure.

Suppose a GW that passes through a ring of test particles and first assume that A(tt)xx 6= 0 ;

A(tt)xy = 0, and then assume the opposite case; with this, we shall be able to see the effect

of each of the independent amplitude components of the wave. In Fig. 1.1 we can observethe time evolution of the system in each case, the former usually known as ’+’ polarizationand the latter as ’×’ polarization. Hence, in general any GW propagating along the z-axiscan be expressed as a linear combination of two independent polarization states,

h = h+e+ + h×e× , (1.7)

where h+ and h× are the two independent components of the GW, and e+ and e× are the


polarization tensors

e+ ≡

0 0 0 00 1 0 00 0 −1 00 0 0 0

; e× ≡

0 0 0 00 0 1 00 1 0 00 0 0 0

. (1.8)

We can see from the panels in Fig. 1.1 that the distortion produced by a gravitationalwave is quadrupolar. Moreover, at any instant, a GW is invariant under a rotation of 180

about its direction of propagation4 and the two independent polarization states can beseen as inclined 45 with respect each other. Putting all pieces together5 is consistent withthe fact that a hypothetical graviton particle (which it is, as yet undiscovered, since we donot have a fully developed theory of quantum gravity) must be a spin S = 2 particle.

1.1.4 Production of gravitational waves

In order to study the propagation of GWs in the vacuum, we have made use of the weakfield approximation, besides assuming the energy momentum tensor to be null; and underthese assumptions we have been able to find an analytical expression for the propagation ofa metric perturbation. Difficulties appear when one is interested in studying the productionof GWs, since now one has to describe the metric close to the compact source, where theweak field approximation is not valid. In the best scenario, it will not be necessary tosolve the full Einstein’s equations, instead it could be enough by making post-Newtonian(PN) approximations; however, in a number of problems one will have to solve Einstein’sequations numerically.

A crude estimation of orders of magnitude and dominant contributions to the luminosityemitted by a source as gravitational radiation, can be done [13] by drawing analogies withthe formulae that describe electromagnetic radiation in terms of the multipolar expansion,but replacing e2 ↔ −m2, in order to go from electrostatic’s Coulomb force to Newton’slaw6.

• In electromagnetic theory, the dominant form of radiation from a moving charge isthe electric dipole radiation, whose luminosity (or “flux”) is given by

Felectric dipole = (2/3) d2 ,

where d is the dipole moment and the dots denote time derivatives. The gravitationalanalogue of the electric dipole moment is the mass dipole moment, summed over adistribution of particles, Ai

d =∑a

maxa .

4By contrast, an electromagnetic wave is invariant under a rotation of 360, and a neutrino wave isinvariant under a rotation of 720.

5In general, the classical radiation field of a particle of spin, S, is invariant under a rotation of 360/S,besides that the different polarization states are inclined to each other at an angle of 90/S.

6This procedure treats gravity as vectorial field instead of a tensorial fields, but it is good enough forour present purposes.

1.2. Astronomical gravitational wave sources 9

Now, we realize that the first derivative of the mass dipole moment is the total linearmomentum of the system, d = p. Since the linear momentum is conserved, it followsthat there can be no mass dipole radiation from any source.

• The next strongest types of electromagnetic radiation are magnetic dipole and electricquadrupole radiation. The magnetic dipole radiation is proportional to the secondtime derivative of the magnetic dipole, µ. As before, its gravitational analogue alsocorresponds to a preserved quantity, in this case the total angular momentum

µ =∑a

ra × (mva) = J ,

hence, there is no radiation either; in other words, there can be no dipole radiationof any sort from a gravitational source.

• In order to find the first not null contribution to gravitational radiation, one mustconsider the quadrupole term. The emitted luminosity predicted by electromagnetismis

Felectric quadrupole =1

20

...Qjk

...Qjk ,

Qjk ≡∑a

ea

(xajxak −

1

3δjkr

2a

).

And the gravitational analogue,

Fmass quadrupole =1

5

⟨ ...J (TT )jk

...J (TT )jk

⟩, (1.9)

J (TT )jk ≡

∑a

ma

(xajxak −

1

3δjkr

2a

)=

∫ρ

(xjxk −

1

3δjkr

2

)d3x , (1.10)

where J (TT )jk is known as the reduced quadrupole moment, the factor 1/5 that appears

instead of the 1/20 is due to the tensorial nature of the gravitational field and “〈·〉”denotes the average over several periods of the source.

We see from these results that a perfectly axisymmetric object rotating around its sym-metry axis will not emit GWs, as its quadrupole moment is constant in time. Only objectswith some sort of axial asymmetry (even if it is small) will produce gravitational radiation.This means that not all the objects in the Universe are candidates to be GW sources, onlystellar-core collapses, CBCs, rapidly spinning NSs, the Big Bang. . . (see Fig. 1.2). Weshall discuss more about astronomical GW sources in Sec. 1.2.

1.2 Astronomical gravitational wave sources

From the wide variety of objects that we find in the Universe, it will be good candi-dates only those ones being compact, rapidly moving and presenting some sort of axialasymmetry (see Sec. 1.1.4). This provides a wide range of possible sources, from NSs(M . 2M and R ∼ RT ) with very short orbital periods, up to galaxy mergers or eventhe Big Bang. In order to estimate the emission frequency of such objects, we can assume


High frequency sources (ground-based detectors)

Stellar core collapses Stellar-mass binary(SNs, newborn BH/NS) Pulsars coalescences Stochastic background

Low frequency sources (space-based detectors)

Supermassive blackGalactic binaries EMRIs hole coalescences Stochastic background

Figure 1.2: Artistic representation of some of the expected GW sources in the high and low frequencybands. [Images: various sources]

that a gravitational-wave (compact) source of mass M cannot be much smaller than itsgravitational radius, 2GM/c2, and cannot emit strongly at periods much smaller than thelight-travel time 4πGM/c3 around this gravitational radius. Correspondingly, the frequen-cies at which it emits are [8]

f .1

4πGM/c3∼ 104 Hz

MM

. (1.11)

To achieve a size of order its gravitational radius and thereby emit near this maxi-mum frequency, an object presumably must be heavier than the Chandrasekhar limit,i.e. M & 1.44M. Thus, the highest frequency expected for strong GWs is fmax ∼ 104 Hz.This define the upper edge of the high-frequency GW band, which spans up to frequencies20 orders of magnitude smaller than fmax. So, this results into a very wide frequency range,that is usually divided into different bands depending on the kind of sources and detectionmethods used in each case. For instance, Ref. [8] distinguish between the high-frequencyband (f ∼ 104 − 1 Hz), the low-frequency band (f ∼ 1− 10−4 Hz), the very-low-frequencyband (f ∼ 10−7−10−9 Hz) and the extremely-low-frequency band (f ∼ 10−15−10−18 Hz). InFig. 1.2 it is shown some of the expected sources in the high-frequency and low-frequencybands.

High-Frequency band, 1− 104 Hz

The high-frequency band is the domain of Earth-based GW detectors: laser interferometersand resonant mass antennas. At frequencies below about 1 Hz, the noise produced eitherby fluctuating Newtonian gravity gradients (e.g. due to inhomogeneities in the Earth’satmosphere which move overhead with the wind) or by Earth seismic vibrations (which

1.2. Astronomical gravitational wave sources 11

are extremely difficult to filter our mechanically below ∼ Hz) become very important. Inorder to detect waves below this frequency, one must fly the detectors into space.

A number of interesting GW sources fall in the high-frequency band, mainly objects withmasses similar to the Sun:

• the stellar collapse to a NS or BH in our Galaxy (or galaxies nearby). Sometimes,they trigger supernovae, in which case one could also observe their electromagneticcounter-parts;

• the rotation and vibration of NSs (pulsars) in our Galaxy;

• the coalescence of NSs and stellar-mass BHs (i.e. M < 1000M) binaries in distantgalaxies;

• stochastic background generated by vibrating loops of cosmic strings, phase transi-tions in the early Universe, or even the Big Bang.

Low-Frequency band, 10−4 − 1 Hz

The low-frequency band is the domain of the detectors flown in space, either in Earthorbit or in interplanetary orbit. In the 1970s, NASA tried to measure the GW effects byDoppler tracking a spacecraft via microwave signals sent from Earth to the spacecraftand there transponded back to Earth, although they did not success. Currently, a jointNASA/ESA project is under development in order to send the Laser Interferometer SpaceAntenna (LISA) into space over the next decade (2020+). As we shall see, LISA will beable to observe a wide variety of sources, some of them located at very high redshifts.Many of the studies performed during this thesis are related to LISA observations.

The ∼ 10−4 Hz lower edge of this frequency band is defined by expected severe difficultiesat lower frequencies in isolating the spacecraft from the buffeting forces of fluctuating solarradiation pressure, solar wind and cosmic rays.

The low-frequency band should be populated by GWs from

• short-period binary stars in our own Galaxy, such as main-sequence stars, white-dwarfs, NSs. . . ;

• white dwarfs, NSs and small BHs inspiraling into massive BHs (M ∼ 3 × 105−3× 107M), so-called extreme mass ratio inspirals (EMRIs), in distant galaxies;

• SMBHs coalescences (M ∼ 105 − 109M);

• it is also expected to observe a low-frequency component of the stochastic backgroundradiation generated in the early Universe.

Very-Low-Frequency band, 10−9 − 10−7 Hz

In order to detect such very-low-frequency signals, it would be necessary to build detectorslarger than the Solar System, what makes to think in different detection techniques. In


particular, Joseph Taylor and others suggested to accurately measure the evolution of theemitting period of several known pulsars located in our own Galaxy over several decades:the idea behind is that when a GW passes over the Earth, it perturbs our rate of flow oftime and thence the ticking rates of our clocks relative to the clocks outside the wave. Thedeviations between the predicted and the observed time of arrivals (TOAs) are known asthe pulsar ‘timing residuals’ and indicate unmodelled effects. In particular, GW signals arenot included in the pulsar timing model, hence, if these residuals are seen simultaneouslyin the timing of several different pulsars, then the cause could well be GWs bathing theEarth.

Unfortunately, the expected signal induced by GWs is small, with typical residuals being< 100 ns. The TOA precision achievable for the majority of pulsars is ∼ 1 ms and mostpulsars show long-term timing irregularities that would make the detection of the expectedGW signal difficult or impossible [14]. However, a sub-set of the pulsar population, themillisecond pulsars, have very high spin rates, much smaller timing irregularities and canbe observed with much greater TOA precision (∼ 30 ns).

The recently created International Pulsar Timing Array project [15] combines observationsof millisecond pulsars from both northern and southern hemisphere observatories with themain aim of detecting GWs in this very-low-frequency band. Given the current theoreticalmodels, it is likely that these GWs will be detected by pulsar timing experiments within 5−10 years. The first detections are expected to be of an isotropic, stochastic GW backgroundcreated by coalescing SMBH systems [16, 17].

Extremely-Low-Frequency band, 10−18 − 10−15 Hz

GWs in the extremely-low-frequency band should produce anisotropies in the cosmic mi-crowave background radiation. Thus, these GW signals could be (indirectly) detected bystudying the cosmic microwave background radiation maps obtained with missions likeCOBE7, WMAP8 and the on-going PLANCK9.

1.3 Direct detection of gravitational waves (from High- andLow-Frequency bands)

It is now 50 years since J. Weber initiated his pioneering development of GW detectors[18] and 40 years since R. Forward [19] and R. Weiss [20] initiated work on interferometricdetectors. Since then, hundreds of experimental physicists have worked to improve thesensitivities of these instruments, at the same time that theoretical physicists have exploredin detail which GR predictions will be able to be tested with these detectors. The currentsensitivities achieved by the interferometric detectors, place GW science on the verge ofdirect observation of the waves predicted by Einstein’s theory of GR and opening theexciting new field of GW astronomy. It is hoped that the first direct observation of GWswill be made in the next few years.

7http://lambda.gsfc.nasa.gov/product/cobe/8http://wmap.gsfc.nasa.gov/9www.esa.int/planck/

1.3. Direct detection of gravitational waves 13

The aim of this section is to give a very brief introduction to the most relevant properties ofthe two most important kind of GW detectors built up to now (cryogenic resonant-massdetectors and interferometric detectors, these last ones both ground- and space-based),which are their main noise sources, what are their current sensitivities, future plans. . .

1.3.1 Cryogenic resonant-mass detectors

Following J. Weber’s efforts in the 1960s and after many years of investigations, in 1986three cryogenic antennas in Stanford, Baton Rouge (ALLEGRO) and at CERN in Geneva(EXPLORER, built by the University of Rome) operated in coincidence for three months;and for increasingly longer periods in the following years. Unfortunately, the Stanfordgroup withdrew from the field, after the untimely death of W. Fairbanks and as a conse-quence of the damage of their detector during the earthquake of 17 October 1989. However,two other groups joined the search, the University of Perth in Australia (NIOBI) and theUniversity of Padova in Italy (AURIGA); and the Rome group designed and assembleda new generation of resonant detector (NAUTILUS) that used a 3He − 4He dilution re-frigerator to cool down the 2500 kg aluminum bar to 0.1 K. Thus, in the 1990s and upto 2006, there was five such resonant-mass instruments around the world [21] working incooperation in the so-called, IGEC10 (International Gravitational Event Collaboration).Currently, both AURIGA and NAUTILUS antennas are still in operation, permanently onwatch in case a supernovae event in our galaxy is detected, covering with their observationsthe periods when all other gravitational wave antennas on the world are offline.

A resonant-mass detector (colloquially known as bar) consists of a solid body, usually acylinder (though spheres are also being considered) of about 3 meters long and a few tons ofweight that vibrates “considerably” when a GW with a frequency similar to its resonancepasses through. The resonant mass is typically made from an alloy of aluminum, butsome have been made of niobium or single-crystal silicon or sapphire. To control thermalnoise, all the resonant-mass detectors are cryogenically cooled down to temperatures, T ∼0.1−6 K, depending on the detector. For instance, the ultra-cryogenic detector NAUTILUSshowed a capability of reaching a temperature as low as 0.1 K being equipped with a3He− 4He dilution refrigerator [22].

In comparison to the interferometric detectors, current resonant-mass detectors are char-acterized to be sensitive in a narrow frequency band (a bandwidth of about 30 Hz) in theuppermost reaches of the high frequency band ∼ 103 to 104 Hz, where photon shot noisedebilitates the performance of interferometric detectors. Thus, resonant-mass detectorshave observational frequency windows complementary to the interferometric GW detec-tors. Current detectors present spectral strain sensitivities of h ∼ 3× 10−21 Hz−1/2, whichcorresponds to a conventional (1 ms) amplitude of GW bursts h ∼ 4× 10−19.

Future detectors may reach sensitivities up to h ∼ 10−21, which corresponds to the quan-tum limit directly obtained from the Heisenberg uncertainty principle. This lower limit inthe sensitivity of the resonant-mass detectors is what propitiated, already 30 years ago,that most of the efforts in designing and building GW detectors were put on the inter-ferometers, which have this lower bound much below. Anyway, resonant-mass detectorshave their importance even in the LIGO/Virgo era, since they provide the opportunity to

10http://igec.lnl.infn.it


Figure 1.3: Basic diagram of an interferomet-ric detector: four masses (test masses) isolatedfrom external vibrations, two of them located closetogether in the vertex of the ’L’ shaped struc-ture and the other two at the end of each ofthe interferometer’s arms. By studying via inter-ferometry techniques, the dephasing between thelaser beams that have traveled along each of thearms, we can obtain precise measures of how thelength difference between the two arms changesover time. In practice, these “masses” are mirrorswith a reflectivity (for the frequency of the laserbeam) close to 1. [Image: LIGO]

search for GW signals in very high frequency regions, where the interferometer detectorsare not that sensitive.

1.3.2 Interferometric gravitational wave detectors

Given the quadrupolar nature of the gravitational waves (see Fig. 1.1), the best approach todirectly detect such waves would be to monitor the time dependency of the length differencebetween two orthogonal directions. With this, since the relative length change producedby a GW in each direction is ∆L

L = h2 ; when the GW plane is aligned with the detector’s

plane, we will have Lx = L + ∆L and Ly = L −∆L, hence ∆LarmsL ≡ Lx−Ly

L = 2∆LL = h.

Moreover, the longer the distance L we are measuring length differences over is, the higherthe sensitivity in measuring GWs, for a fixed ∆L precision, will be.

These arguments made scientists think about the Michelson-Morley interferometer as thestarting design to built alternative (to resonant-mass bars) and more sensitive GW de-tectors. The basic diagram of an interferometric (ground-based) detector consists in fourmasses (test masses) suspended as pendula using seismic isolation systems, distributedas it is shown in Fig. 1.3 and the corresponding optical system to monitor the separa-tion between masses. Two masses are located very close together, in the vertex of the ‘L’shaped structure, whereas the other two are at the end of each interferometer’s arms11.In ground-based detectors, these “masses” are mirrors with an extremely high reflectivityand therefore they are already part of the optical system.

To measure the relative length difference of the arms, a single laser beam is split at theintersection of the two arms (beam splitter mirror). Half of the laser light is transmittedinto one arm while the other half is reflected into the second arm. Laser light in each armbounces back and forth between these mirrors (test masses) forming a Fabry-Perot cavity,and finally returns to the intersection, where it interferes with light from the other arm. Ifthe lengths of both arms have remained unchanged, then the two combining light wavesshould completely subtract each other (destructively interfere) and there will be no lightobserved at the output of the detector (photodetector). However, if a GW were to slightly[assuming h ∼ 10−21 and L = 4 km, then ∆L ∼ 2 × 10−18 m ∼ 1/1000 the diameter ofa proton] stretch one arm and compress the other, the two light beams would no longer

11The arms of the interferometer do not have to be orthogonal, although this is the optimal configurationand indeed, the one used in all ground-based detectors. However, the future Laser Interferometer SpaceAntenna, for design reasons, will have 60 arms and it will consist in 6 free masses, instead of 4.


completely subtract each other, yielding light patterns at the detector output. Encoded inthese light patterns is the information about the relative length change between the twoarms, which in turn tells us about what produced the GWs.

Many things on Earth are constantly causing very small relative length changes in thearms of the interferometers. These every-present terrestrial signals are regarded as noise.The goal when designing a GW detector is to reduce this noise contributions to levelsbelow the expected GW amplitudes and current technology is on the verge of reachingthese levels.

• One of the most important parts in any ground-based GW detector are the seismicisolation systems. On one hand, there are tiny magnets attached to the back of eachmirror, and the positions of these magnets are sensed by the shadows they cast fromLED light sources. If the mirrors are moving too much, an electromagnet creates acountering magnetic field to push or pull the magnets and mirror back into position[this method is not only good for countering the motion of the mirrors due to localvibrations, but also to counter the tidal force of the Sun and the Moon]. Externally,there are very sophisticated hydraulic systems that filter out the Earth’s surfacevibrations.

• Also, all the optical components are placed inside a vacuum. On the superficiallevel this keeps air current from disturbing the mirrors, but mainly this is to ensurethat the laser light will travel a straight path in the ∼ kilometer arms [notice thatslight temperature differences across the arm would cause the light to bend due totemperature dependent index of refraction].

• First generation of detectors include an extra mirror between the laser source andthe beam splitter, called power recycling mirror. Its purpose is to coherently re-injectthe fraction of power that the detector is sending back to the laser source and that,otherwise would be wasted. With this, one gets more power into the detector’s armsand reduces some of the noise contributions. Also, GEO-600 has been testing a signalrecycling mirror, which has the same purpose as the former, but this one re-injectsa fraction of the power sent to the photodetector. It is planned that Adv. LIGO isgoing to incorporate this extra mirror.

• A lot of development has also been done in the laser technology. The laser beaminjected into the arms is previously frequency and amplitude stabilized [a changinglaser frequency would add noise in the output of detector], and also they add somesecondary modes necessary to lock the different Fabry-Perot cavities that are formedbetween the mirrors that compose the interferometer.

• Also, there are some on-going investigations towards building cryogenic interfer-ometric GW detectors, e.g. the japanese Large Cryogenic Gravitational Telescope(LCGT)12 or the european 3rd generation GW interferometer project, called EinsteinTelescope (ET).

Current and future interferometric GW detector projects are (see also, Fig. 1.4):

12http://tamago.mtk.nao.ac.jp/spacetime/lcgt e.html


LIGO Hanford observatory Einstein Telescope (ET) LISA

Figure 1.4: Pictures (or artistic representations) of some interferometric GW detectors. [Images: institu-tional web pages]

• The Laser Interferometer Gravitational-wave Observatory (LIGO) consists in threedetectors, two colocated interferometers (with 2 km and 4 km arms, respectively) inHanford (WA) and another 4 km detector in Livingston (LA). Currently they are themost sensitive GW detectors in the world and they were operating in science modeup to Oct. 2010, when it concluded the sixth science mode run, called ‘S6’. Sincethat date, all scientists and engineers are working on the implementation of whatwill be the second generation of GW interferometric detectors, so-called Adv. LIGO.Some of the enhancements of this new generation (like the output mode cleaner andhigher power mode) have been already tested during this very last science run, butnow much greater updates are being implemented. Adv. LIGO is expected to beoperational in about 4 years, with an improvement of almost an order of magnitudein sensitivity, i.e. a factor 1000 in observable volume.

• The Virgo detector is the result of a french-italian collaboration and consists in a 3 kminterferometer located in Cascina (Italy). In general, the sensitivity of this detector isa little bit worse than LIGO; however it has a more sophisticated suspension systemthat provides better sensitivity at lower frequencies, where the inspiral signals fromCBCs within the Virgo supercluster are located. During the last decade, the networkLIGO-Virgo have been the most sensitive GW detectors in the world, and they havetried to collaborate and have ‘science mode’ time in coincidence. Indeed, some yearsago the two scientific collaborations joined efforts creating the LIGO-Virgo ScientificCollaboration. As in LIGO, Virgo is currently being upgraded to Adv. Virgo; alsoan order of magnitude of improvement in sensitivity is expected from this secondgeneration detector.

• GEO-600 is a 600 m detector built as a collaboration between the United Kingdomand Germany. The detector is located in Hannover (Germany) and, although it cannot compete with LIGO or Virgo, it has been used as a test bank for the technology[mainly in collaboration with the LIGO Scientific Collaboration (LSC)] that, then,is going to be implemented in the next generation of detectors. For instance, GEO-600 has been running with a high power laser, output mode cleaner, signal recyclingmirror, monolithic suspensions and electrostatic test mass actuators [23]; that noware planned to be installed in Adv. LIGO and Adv. Virgo. The future plans forthis detector is to improve its sensitivity curve with further experimental upgradesin what will be GEO-HF, such as the tuned signal recycling and DC readout, theimplementation of an output mode cleaner, the injection of squeezed vacuum into theanti-symmetric port, the reduction of the signal recycling mirror reflectivity and apower increase. In comparison with its contemporary detectors (Adv. LIGO and Adv.


Virgo), GEO-HF will be focused in improving its sensitivity in the high frequencyrange.

• TAMA-300 is a 300 m interferometric japanese project built in the city of Tokyo. Formany years it was the most sensitive GW detector in the world, but now it is usedjust as a bank test for the extremely ambitious project of building an underground,cryogenic GW detector, the LCGT project.

• AIGO is an australian prototype for a GW detector. Currently, the most likely nextstep will be to build an exact replica of one of the LIGO interferometers in Australia.This is the cheapest solution and adding these new detector to the current networkwould significantly increase the network’s sky resolution.

• Einstein Telescope (ET)13 is the european project to build a third generation GWdetector. GEO and Virgo members [and other european institutions] have joinedefforts in designing what should be the new generation of GW interferometric detec-tors, possibly underground and using cryogenic techniques. Currently, ET is in itsdesign phase, but it is receiving a lot of attention from all the european GW scientificcommunity. Its design sensitivity should be an extra order of magnitude below theadvanced detectors noise curves.

• The Laser Interferometer Space Antenna (LISA) is the most ambitious project tobuilt an space-based interferometric GW detector to observe the Low-Frequencyband. The technology needed for this project is so demanding, that the two mostimportant space agencies, NASA and ESA, have joined their efforts to built LISA.It will consist in a constellation of 3 free-falling spacecrafts forming an equilateraltriangle of 5× 106 km long each edge, and following the Earth in its motion aroundthe Sun, about 20 behind. The orbits have been designed in such a way that thetriangle constellation will also rotate 360 around its center every time it completesan orbital cycle around the Sun (i.e. every year), by doing this, the detector obtainsmore angular resolution. Although the main working principle of LISA is the sameas in the ground-based interferometers, the length of its arms (≈ 16.7 light-seconds),the fact that each spacecraft can have its own independent movement besides theeffect of GWs and the impossibility of repairing anything once it has been launchedinto space, make LISA’s design much more complicated. The expected launch dateof LISA is in 2020+, although in order to test the feasibility of its technology aprecursor mission fully managed and funded by European Space Agency (ESA),LISA Pathfinder, is being launched in 2012.

• As a follow-on mission to LISA, the Big Bang Observer (BBO) is proposed to NASAas a Beyond Einstein mission, targeted at detecting stochastic gravitational wavesfrom the very early universe in the band 0.03 Hz to 3 Hz [7]. BBO will also be sensitiveto the final year of binary compact body (NSs and stellar-mass BHs) inspirals outto z < 8, mergers of intermediate mass black holes at any redshift, rapidly rotatingwhite dwarf, explosions from Type 1a supernovas at distances less than 1 Mpc and< 1 Hz pulsars.

• While BBO is seen as a successor to LISA in the US and Europe, the DECi-hertzInterferometer Gravitational wave Observatory (DECIGO) is the future Japanesespace gravitational wave antenna [7]. The goal of DECIGO is to detect GWs from

13http://www.et-gw.eu/


various kinds of sources mainly between 0.1 Hz and 10 Hz. The current plan isfor DECIGO to be a factor 2-3 less sensitive than BBO, but for an earlier launch.DECIGO can play a role of follow-up for LISA by observing inspiral sources thathave moved above the LISA band, and can also play a role of predictor for terrestrialdetectors by observing inspiral sources that have not yet moved into the terrestrialdetector band. In order to increase the technical feasibility of DECIGO before itsplanned launch in 2024, the Japanese are planning to launch two milestone missions:DECIGO pathfinder and pre-DECIGO.

An important difference of the GW detectors in comparison to standard electromagnetictelescopes is that the former are omnidirectional, i.e. with a single strain time series h(t),we observe overlapped signals coming from any direction in the sky. The advantage is thatwe are always making all-sky observations and therefore we will not miss any event becauseof ‘not pointing’ at the right sky location. However, this also implies some disadvantages,in particular, the sky resolution of GW detectors will be very poor compared to anyelectromagnetic telescope; and also it can be a problem to observe too many overlappedsignals, which can even be indistinguishable becoming an extra source of noise (e.g. thisis case of galactic binaries with the LISA detector).

The angular resolution in GW detectors is basically given by three effects: (i) the time-delay in observing an event from distant locations (this implies having several detectorsworking in coincidence); (ii) the Doppler effect14 due to the relative motion of the detectorwith the respect to the source and (iii) the anisotropic sensitivity of the GW detectors(see, for instance, Fig. 2.2). Any of these effects require either long observation times sothat the velocity and orientation of the detector have substantially changed during theobservation, or the simultaneous observation of the same event with several detectors wellseparated one from each other; a ‘burst’ signal observed by a single GW detector will implyan unknown sky location. For these reasons, it has been built a network of ground-baseddetectors located all over the world that work collaboratively in order to accumulate themost coincidence observational time as possible. For the case of LISA, most of the expectedsources are long-lived sources in the LISA band and therefore, the yearly motion of thedetector around the Sun will provide a good sky resolution. For instance, in Fig. 1.5we show the same inspiral signal from a 106M − 107M SMBH coalescence at differentlocations, observed by LISA during the last year of inspiral phase; we see how the amplitudemodulations clearly allow one to localize long-lived GW signals. The Doppler shift, on theother hand, can not be measured from signals in the Low-Frequency band since the typicalDoppler frequency shifts, ∆f = f vc , are smaller than the frequency resolution provided bythe observational time, δf = 1/Tobs.

In comparison to the resonant-mass detectors, the interferometers are wide-band detectors.In particular, the km-scale ground-based detectors will cover the High-Frequency band(1− 104 Hz) of GW sources, whereas the space missions (such as LISA) will be observingsignals from the Low-Frequency band (10−4−10−1 Hz) of the GW spectrum. The ground-based detectors that have been in operation up to this date (first generation), have reachedspectral sensitivities15 of

√Sn ∼ 3 × 10−23 Hz−1/2, and if one considers the LIGO-Virgo

network, the expected detection rates for NS-NS systems is 0.02 yr−1 [24]. Currently, the

14Actually, given the frequency resolution and typical velocities of the detectors, only the signals in theHigh-Frequency band will produce measurable frequency shits, since ∆f = f v

c.

15See a proper definition of the noise power spectral density in Sec. 3.1.

1.4. Compact Binary Coalescences 19

−350 −300 −250 −200 −150 −100 −50 0−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

−18

time (day)

h(t

)

cos θ

N = −0.6 ; φ

N = 1

cos θN

= 0.8 ; φN

= 2

cos θN

= 0.08 ; φN

= 1.5

Figure 1.5: Time-domain waveforms observedby LISA from a SMBH binary system of 106M−107M at redshift z = 1 during the last year ofthe inspiral phase before the last stable orbit (in-stant of time where we set t = 0). The three dif-ferent curves represent the same source located atdifferent position in the sky, which clearly showsthe effect of the amplitude modulation in the ob-served signals due to the anisotropic sensitivity ofthe LISA detector. [Plot: own production]

new generation of ground-based detectors is being implemented and we expect them tobe operational by 2015; the expected spectral sensitivities of such advanced detectors is√Sn ∼ 4× 10−24 Hz−1/2 [24], i.e. a factor almost 10 of improvement in comparison with

the first generation, which directly translates into a factor 10 of increase in the horizondistance and therefore a factor 103 in ‘observable volume’. The (likely) expected eventsrates for the Advanced LIGO-Virgo network for the same NS-NS systems is 40 yr−1 [24].Finally, let us say that a third generation of GW interferometric is current in the designingphase, and for instance, the expected sensitivity for ET is

√Sn ∼ 3× 10−25 Hz−1/2 [25].

1.4 Compact Binary Coalescences

Among the different potential sources for interferometric GW detectors, compact binarycoalescences (CBCs) in the Universe are the most promising ones, both in terms of scientificresults and detectability, given the wide range of masses and distances that can be observed.Ground-based detectors will observe NSs and stellar-mass BHs binary mergers (i.e.: the lastphase of the coalescence) at any point of the Virgo Supercluster, including the Local Groupand therefore the Milky Way. On the other hand, space-based detectors are sensitive to alower frequency band and therefore will observe mergers of SMBHs (more than 105 M)at any place in the Universe; besides tens of thousands of stellar-mass binaries located inour Galaxy that are in an early stage of their coalescing evolution, slowly inspiraling onearound each other. See Sec. 3.5 for further details.

In order to detect all the signal-to-noise ratio (SNR) and to extract reliable physicalinformation from GW observations, an accurate representation of the dynamics of thesource and its emitted gravitational waveform is needed. The coalescence of a compactbinary can be divided into three successive phases according to its dynamics: (i) an initialadiabatic inspiral phase where the emitted gravitational waveform is a chirping signal (i.e.:frequency and amplitude slowly increasing over time) that can be analytically described asa PN expansion; (ii) when the two objects are very close together, the velocities becomerelativistic, the energy emitted importantly increases and the process is not adiabaticanymore (merger phase); thus, full GR equations must be solved in order to provide afaithful representation of the dynamics; finally, after the merger, (iii) the resulting excitedKerr BH will settle down (ring-down phase) by emitting gravitational radiation analyticallydescribed as a superposition of quasi-normal modes.


It is known that the extremely long inspiral process that precedes the ‘visible’ (mostenergetic) final stages of the evolution (late inspiral, merger and ring-down), tends tocircularize the orbits [26, 27]. This is the reason why one normally expects to find circular orquasi-circular orbits in Nature. Only recent catastrophic events near the CBC, three-bodyinteractions or highly spinning compact objects can break this quasi-circular condition[28–30]. In this thesis, we shall always assume circular orbits.

Also, we shall consider the particular case of non-spinning black holes (BHs) [or compactstarts with negligible effects coming from their equation of state]. Although this representsa restriction with respect to what is expected to be found in the Universe, the current ‘stateof the art’ on source modeling is well-stablished for non-spinning BHs but it just startsbeing developed for spinning compact objects with possible presence of matter. Some ofour future plans are related to the extension of the analyses performed in this thesis tothe more general case of spinning BHs or NSs with different equations of state.

References 21

References

[1] R. A. Hulse, J. H. Taylor, Astrophys. J. 195 L51-L53 (1975).

[2] J. M. Weisberg, J. H. Taylor, Phys. Rev. Lett. 52 1348-1350 (1984).

[3] J. H. Taylor, J. M. Weisberg, Astrophys. J. 345 434-450 (1989).

[4] C. M. Will, Theory and Experiment in Gravitational Physics, (Cambridge Univ. Press) (1994).

[5] T. Damour, J. H. Taylor, Phys. Rev. D 45 1840-1868 (1992).

[6] T. Damour, B. R. Iyer, B. S. Sathyaprakash, Phys. Rev. D 63 044023 (2001) [Erratum-ibid.D 72 029902 (2005)].

[7] J. Marx et al., Gravitational Waves International Committee Roadmap, The future of gravi-tational waves astronomy, http://gwic.ligo.org/roadmap/Roadmap 100627.pdf (2010).

[8] K. S. Thorne, Gravitational waves, Proceedings of the Summer Study on particle and nu-clear astrophysics and cosmology in the next millennium, Snowmass 1994:0160-184, Editedby E. W. Kolb and R. D. Peccei (River Edge, N.J., World Scientific) (1995), [gr-qc/9506086].

[9] C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, (W. H. Freeman and Co.) (1973).

[10] B. F. Schutz, A First Course in General Relativity, Second Edition, (Cambridge UniversityPress, Cambridge, England) (2009).

[11] K. S. Thorne, in 300 Years of Gravitation, edited by S. W. Hawking and W. Israel (CambridgeUniversity Press, Cambridge, England) (1987) p. 330.

[12] B. F. Schutz, Gravitational wave astronomy, [gr-qc/9911034].

[13] M. Hendry, An Introduction to General Relativity, Gravitational Waves and Detection Prin-ciples, Second VESF School on Gravitational Waves, Cascina, Italy (2007).

[14] G. Hobbs, A. Lyne, M. Kramer, Chin. J. Astron. Astrophys. 6 169 (2006).

[15] G. Hobbs et al., Class. Quant. Grav. 27 084013 (2010).

[16] A. Sesana, A. Vecchio, C. N. Colacino, Mon. Not. R. Astron. Soc. 390 192 (2008).

[17] A. Sesana, A. Vecchio, M. Volonteri, Mon. Not. R. Astron. Soc. 394 2255 (2009).

[18] J. Weber, Phys. Rev. 117 306 (1960).

[19] G. E. Moss, L. R. Miller, R. L. Forward, Applied Optics 10 2495 (1971).

[20] R. Weiss, Quarterly Progress Report of RLE, MIT 105 54 (1972).

[21] P. Rapagnani, Class. Quant. Grav. 27 194001 (2010).

[22] P. Astone et al., Class. Quant. Grav. 25 114048 (2008).

[23] H. Grote [for the LIGO Scientific Collaboration], Class. Quant. Grav. 27 084003 (2010).

[24] J. Abadie et al. [LIGO and Virgo Collaborations], Class. Quant. Grav. 27 173001 (2010).

[25] S. Hild, S. Chelkowski, A. Freise, “Pushing towards the ET sensitivity using ’conventional’technology,” [arXiv:0810.0604].

[26] P. C. Peters, J. Mathews, Phys. Rev. 131 435-439 (1963).

[27] P. C. Peters, Phys. Rev. 136 B1224-B1232 (1964).


[28] K. Gultekin, M. Coleman Miller, D. P. Hamilton, Astrophys. J. 640 156-166 (2006).

[29] M. Campanelli, C. O. Lousto, Y. Zlochower, Phys. Rev. D 77 101501 (2008).

[30] A. Sesana, Astrophys. J. 719 851-864 (2010).

Chapter2Observed gravitational signals:from source emission to detectedsignals

From the source simulation point of view, GWs are small perturbations of the surroundingmetric space that move away from the source, traveling at the speed of light. Along itsway between the source and the observer, the intensity carried by the GWs will obey aninverse-square law (the emitted wavefront can be considered to be spherical) and, as withall radiation fields, the amplitude of the GWs falls off as D−1

L far from the source [1]. Oncethe gravitational wavefront reaches the observer (in our case an interferometric detector),it modifies the metric of the surrounding space producing changes in the relative lengthdifference of the two interferometer arms, which can be measured with extremely goodaccuracy making use of interferometric techniques.

The conversion process between the metric perturbation in the source frame and themeasured wave-induced relative length changes of the two arms, despite being based onsimple and well stablished spherical harmonics decomposition, coordinates transformationsand matrix projections; it contains some subtleties and convention elections that may berelevant for any work related with GW data analysis. For this reason, we consider veryimportant to, at least once, carefully study and fully understand the whole process frombeginning to end. Indeed, this is the main purpose of the present chapter, where we willconsider the particular case of CBCs.

Following this idea, this chapter has been organized in four sections, where we shall startfrom the raw output of a numerical simulation, this is, the Regge-Wheeler and Zerilli

functions, Ψ(o)`m and Ψ

(e)`m; in order to then, add their corresponding angular dependence

in form of spin-weighted spherical harmonics (see Sec. 2.1); followed by their projectioninto the detector’s arms (Sec. 2.2). Since we are interested in data analysis studies, weshall write the final expression for the measured GW strain in its most compact form, asa simple superposition of (co)sinusoidal functions of the different GW phase harmonics

23

24 Chapter 2: Observed GW signals: from emission to detected signals

(Sec. 2.3). The following diagram schematizes the derivation process that we shall follow:Ψ

(o)`m,Ψ

(e)`m

Sec. 2.1−−−−−−→ h+, h× Sec. 2.2−−−−−−→ h(t) [formal]

Sec. 2.3−−−−−−→ h(t) [useful] .

Finally, in Sec. 2.4 we shall study some residual gauge-freedom in the measured GW strain,h(t), that may create (exact or approximated) degeneracies between different regions inthe parameter space. If they are exact, we should be able to restrict the parameter spaceand avoid exploring equivalent regions [this is indeed very useful in ‘blind’ searches, whereone has to explore the whole parameter space]; in case they are merely approximated orvalid in limit cases, then we won’t be able to reduce the parameter space volume, but,in any case, the knowledge about their existence will help us to increase the efficiency ofMarkov chain Monte Carlo searches (see Chapter 6).

Throughout this chapter, we have been very careful trying to always write the explicitdependencies of the different quantities as they appear during the derivation process. Ifa certain quantity is time-dependent, it is denoted by a t or τ , depending on whetherone is talking about the coordinate time t or the proper time at the source’s referenceframe τ , followed by a semicolon and the list of physical parameters it depends on,e.g. an = an(τ ;λsrc). This should help the reader to keep track of where the dependencieson different parameters come from.

2.1 Emitted gravitational wave signals from compact bina-ries

We consider a GW signal traveling through the space in the −N direction and generated bya rotating system with angular momentum, L. In this notation, N stands for the unitaryvector from the detector to the source (i.e. representing the source sky location), andL shall represent an unitary vector parallel to L (see Fig. 2.1). Let us denote the anglebetween the angular momentum and the propagation direction of the GW as ι, that canbe univocally obtained by the following two scalar expressions:

cos ι = −L · N , sin ι = |L× N| (2.1)

Notice that sin ι is positive defined and therefore ι ∈ [0, π]. The other spherical polar anglethat univocally determine the −N vector measured in the source reference frame (i.e. thez axis parallel to the angular momentum, L, and x − y containing the orbital plane) isβ, which is measured over the orbital plane from an arbitrary, but fixed, direction in thisframe.

All the potential GW sources considered both for ground-based and space-based detectorsare far enough to neglect parallax effects due to the motion of the detectors around theSun and therefore, N will be constant. On the other hand, fast rotating systems canhave precessing orbital planes in the observational time scale (specially, if one considersspinning compact objects, where the PN coupling between the bodies’ spins and theirorbital angular momentum undergoes Lense-Thirring precession [2, 3]) and in these cases,L would be variable over the observation time. Thus, ι and β may not be constant forsome cases, although they are going to be in all the particular cases considered in thisthesis.

2.1. Emitted gravitational wave signals from compact binaries 25

In general, when solving the dynamics of a CBC, given the spherical symmetry of thebackground metric space, the metric perturbations, h, can be expanded [4, 5] in multipolesof odd or even parity according to their transformation properties under parity [this is,(ι, β)→ (π− ι, π+ β)], leaving all the angular dependency for the spin-weighted sphericalharmonics. Concretely, the emitted GW signal can be written in terms of the Regge-Wheeler and Zerilli functions, Ψ(o) and Ψ(e) respectively, which can be expanded in thefollowing way [4, 5]:

h+ − ih× =1

DL

∞∑`=2

∑m=−`

√(`+ 2)!

(`− 2)!

(Ψ

(e)`m(τ ;λsrc) + iΨ

(o)`m(τ ;λsrc)

)−2Y `m(ι, β)

≡∞∑`=2

∑m=−`

h`m(τ ;λsrc) −2Y`m(ι, β) . (2.2)

In this previous expression, h(+,×) ≡ h(+,×)(τ ;λsrc, ι, β), where λsrc = m1,m2, DL, spins . . .are the source physical parameters and DL the luminosity distance. Moreover, −2Y

`m(ι, β)are the (s = −2) spin-weighted spherical harmonics [5, 6] (see also Sec. 2.1.1) and h`mare simply the spin-weighted spherical harmonic components, as they are defined in [4].Finally, let us notice that the sum over ` starts at ` = 2, because we are only focusedon the radiative degrees of freedom of the perturbation; it turns out that the monopolecomponent of the metric for a vacuum perturbation (` = 0) represents a variation in themass-parameter of the Schwarzschild solution, whereas the dipole component (` = 1) canbe removed either by means of a suitable gauge transformation (when it has even parity) orby introducing an angular momentum onto the background metric (for odd-parity metricperturbations).

2.1.1 Spin (s = −2) weighted spherical harmonics

The analytical expressions to compute the (s = −2) spin-weighted spherical harmonics interms of the scalar spherical harmonics are given, for instance, by Nagar and Rezzolla [5]in terms of the standard scalar spherical harmonics:

−2Y`m(ι, β) ≡

√(`− 2)!

(`+ 2)!

(W `m(ι, β)− iX

`m(ι, β)

sin ι

), (2.3)

where

W `m(ι, β) ≡ ∂2ι Y

`m − cot ι ∂ιY`m − 1

sin2 θ∂2βY

`m (2.4)

X`m(ι, β) ≡ 2(∂2ιβY

`m − cot ι ∂ιY`m), (2.5)

Y `m ≡ Y `m(ι, β) being the standard scalar spherical harmonics and ∂xf(x) representingthe partial derivative of the function f(x) with respect to the variable x. While theseexpressions can easily be reproduced with straightforward algebra, they are tedious toderive and hard to find in the literature; for these reasons we have decided to includetheir explicit expressions in Table 2.1. By combining two symmetry properties of thegeneral spin-weighted spherical harmonics: sY

`m ∗ = (−1)s+m−sY`−m and sY

`m(π− ι, π+β) = (−1)`−sY

`m(ι, β) and fixing s = −2, we can relate the ±m modes through a paritytransformation:

−2Y`−m(ι, β) = (−1)`+m−2Y

`m ∗(π − ι, π + β) , (2.6)


where the asterisk denotes complex conjugate.

For instance, the particular explicit expressions for the (`,m) = (2,±2) spin-weightedspherical harmonics read

−2Y22(ι, β) =

1√4!

ei2β√ 15

32π(3 + cos 2ι)− i

iei2β√

158π sin 2ι

sin ι

=

1

16

√5

πei2β (3 + cos 2ι+ 4 cos ι)

=1

2

√5

πcos4 ι

2ei2β (2.7)

and

−2Y2−2(ι, β) = −2Y

22 ∗(π − ι, π + β) =1

2

√5

πsin4 ι

2e−i2β . (2.8)

The rest of the (s = −2) spin-weighted spherical harmonics (up to ` = 6) are listed inTable 2.1.

2.1.2 General expression for h+ and h×

In this thesis we consider non-spinning compact objects coalescing in near circular orbits.For these systems, Kidder [4] makes use of the fact that the mass [current] multipoles onlycontribute to components with `+m even [odd], together with other symmetry propertiesto determine the relation between the h`m and h`−m components,

h`−m = (−1)`h∗`m . (2.9)

Moreover, one can, in general, write the (spin-weighted) spherical harmonic componentsof the gravitational emission, h`m, as a complex amplitude times a complex exponentialfunction of the (minus) m-th harmonic of the orbital phase [4]. This is,

h`m ≡ (hR`m + ihI`m)e−imφorb (2.10)

h`−m = (−1)`h∗`m = (−1)`(hR`m − ihI`m)e+imφorb , (2.11)

where h(R,I)`m ≡ h

(R,I)`m (τ ;λsrc) are real quantites; φorb ≡ φorb(τ ;λsrc) is the orbital phase

of the binary and we have used Eq. (2.9) to obtain the general expression for h`−m. Werefer the reader to Ref. [4] for the explicit expressions of h`m as a PN expansion up to themaximum known PN order.

Comment on the way to proceed and its alternative

At this point, it is important to notice that we could proceed in two different ways. FromEq. (2.10), we have h`m = (hR`m + ihI`m)e−imφorb , consisting in a complex amplitude thatdepends on the particular (`,m) mode that one is considering, and a real phase independentof ` and with a trivial dependency on m. On the other hand, one might be interested in


` = 2

−2Y22 = 1

2e2iβ√

5π cos

[ι2

]4−2Y

21 = 14eiβ√

5π (1 + cos ι) sin ι

−2Y20 = 1

4

√152π sin2 ι

` = 3

−2Y33 = −e3iβ

√212π cos

[ι2

]5sin[ι2

]−2Y

32 = 12e

2iβ√

7π cos

[ι2

]4(−2 + 3 cos ι)

−2Y31 = − 1

32eiβ√

352π (sin ι− 4 sin 2ι− 3 sin 3ι) −2Y

30 = 14

√1052π cos ι sin2 ι

` = 4

−2Y44 = 3

64e4iβ√

7π csc

[ι2

]4sin6 ι

−2Y43 = −3e3iβ

√72π cos

[ι2

]5 (−2 sin[ι2

]+ sin

[3ι2

])−2Y

42 = 34√πe2iβ cos

[ι2

]4(9− 14 cos ι+ 7 cos 2ι)

−2Y41 = 3

32√2πeiβ(3 sin ι− 2 sin 2ι+ 7(sin 3ι+ sin 4ι))

−2Y40 = 3

16

√52π (5 + 7 cos 2ι) sin2 ι

` = 5

−2Y55 = −e5iβ

√330π cos

[ι2

]7sin[ι2

]3−2Y

54 = e4iβ√

33π cos

[ι2

]6(−2 + 5 cos ι) sin

[ι2

]2−2Y

53 = − 1256e

3iβ√

332π (14 sin ι− 8 sin 2ι+ 13 sin 3ι+ 36 sin 4ι+ 15 sin 5ι)

−2Y52 = 1

8e2iβ√

11π cos

[ι2

]4(−32 + 57 cos ι− 36 cos 2ι+ 15 cos 3ι)

−2Y51 = 1

256eiβ√

77π (−2 sin ι+ 8 sin 2ι− 3 sin 3ι+ 12 sin 4ι+ 15 sin 5ι)

−2Y50 = 1

32

√11552π (5 cos ι+ 3 cos 3ι) sin2 ι

` = 6

−2Y66 = 3

512e6iβ√

715π csc

[ι2

]4sin8 ι

−2Y65 = − 1

2e5iβ√

2145π cos

[ι2

]7(−1 + 3 cos ι) sin

[ι2

]3−2Y

64 = 18e

4iβ√

1952π cos

[ι2

]6(35− 44 cos ι+ 33 cos 2ι) sin

[ι2

]2−2Y

63 = 120483e3iβ

√13π (20 sin ι− 51 sin 2ι+ 6 sin 3ι− 5(4 sin 4ι+ 22 sin 5ι+ 11 sin 6ι))

−2Y62 = 1

256e2iβ√

13π cos

[ι2

]4(1709− 3096 cos ι+ 2340 cos 2ι− 1320 cos 3ι+ 495 cos 4ι)

−2Y61 = 1

1024eiβ√

652π (20 sin ι− 17 sin 2ι+ 54 sin 3ι− 12 sin 4ι+ 66 sin 5ι+ 99 sin 6ι)

−2Y60 = 1

512

√1365π (35 + 60 cos 2ι+ 33 cos 4ι) sin2 ι

Table 2.1: (s = −2) spin-weighted spherical harmonics, −2Y`m ≡ −2Y

`m(ι, β), up to ` = 6, obtained fromthe explicit expression Eq. (2.3). The harmonics with m < 0 can be obtained from these ones, making useof Eq. (2.6).


having a real amplitude and phase values, but adding the (`,m) dependency also to thephase:

h`m = |h`m|e−iε`m , (2.12)

where

|h`m| ≡√

(hR`m)2 + (hI`m)2 , (2.13)

ε`m ≡ mφorb − arctan(hI`mhR`m

). (2.14)

In this thesis we shall proceed using Eq. (2.10) since, indeed, we shall make use of its trivialphase dependency with `,m, but it is important to notice that in a different context, itmight be more useful to proceed with the other generic way (2.12) of writing things.

Dominant (2, 2) mode

By combining Eqs. (2.6) and (2.10)-(2.11) and introducing them into (2.2) we shall obtainbelow the generic expressions for h+ and h×, but before doing this and for pedagogicalreasons, let us start first deriving results for the dominant (` = 2,m = ±2) modes [usuallyreferred to them just as the (2,2) mode]. Thus, we proceed by introducing the explicitexpressions of the spin-weighted spherical harmonics for these modes [Eqs. (2.7)-(2.8) and(2.10)-(2.11)] into Eq. (2.2),

h+ − ih× = h22(τ ;λsrc) −2Y22(ι, β) + h2−2(τ ;λsrc) −2Y

2−2(ι, β)

=1

2

√5

π

[(hR22 + ihI22)e−i(2φorb−2β) cos4 ι

2+ (hR22 − ihI22)ei(2φorb−2β) sin4 ι

2

],

and, after some basic algebra one can finally get

h+(τ ;λsrc, ι, β) =1

2

√5

π

(1 + cos2 ι

2

)[hR22 cos(2φorb − 2β) + hI22 sin(2φorb − 2β)

],

h×(τ ;λsrc, ι, β) =1

2

√5

πcos ι

[hR22 sin(2φorb − 2β)− hI22 cos(2φorb − 2β)

], (2.15)

where we recall the explicit dependencies, h(R,I)22 ≡ h(R,I)

22 (τ ;λsrc) and φorb ≡ φorb(τ ;λsrc).Notice that, since we are considering a m = 2 mode, the GW signal contains the secondharmonic of the orbital phase, always combined with the spherical angle measured overthe orbital plane, β, in the same way. We shall see that this is satisfied in the general caseand, since the (2, 2) mode is the one carrying most of the GW energy1, this is why

φgw(τ ;λsrc, β) ≡ 2φorb(τ ;λsrc)− 2β (2.16)

is usually defined as the GW phase, and the only dependency of the measured GW signalwith the angle β can be absorbed with the arbitrary initial orbital phase, namely φ0 ≡φgw(t = 0), if we assume β to be constant. For this reason, we represent the β-dependencyof φgw as ‘β’. Note in (2.16) the factor 2 with respect to the orbital phase; which shallalso be present when one talks about the GW frequency, F ≡ φgw/(2π), being twice theorbital frequency [when β is constant].

1At least, for non-spinning sources in quasi-circular orbits.


For many data analysis purposes, it is enough to just consider the dominant (2, 2) modeof the spherical harmonic decomposition and, as a matter of fact, one, normally, onlyconsiders the leading order from the PN expansion of h22, obtaining what is usually calledrestricted post-Newtonian approximation. Under this approximation, the GW amplitudeturns out to be,

hR22(τ ;λsrc) = 2

√π

5an ; hI22(τ ;λsrc) = 0

[where the subscript ‘N’ stands for ‘Newtonian’, as the leading order in the PN expansion],whereas the GW phase, φgw, incorporates all the known PN corrections. The use of therestricted PN approximation is justified in a number of cases, since it simplifies significantlythe expression of the gravitational waveform while keeping all the PN corrections to thephase, which is the most sensitive quantity when describing long GW signals, as anydephasing error is accumulated over many cycles. However, as we shall see in Chapter 4,in some cases the inclusion of full information from all the modes may significantly improvethe detection and, specially, the parameter estimation of GW signals. The reason behindthis fact is rather not because of adding PN corrections to the amplitude of the dominantharmonic, but because of adding contributions to higher harmonics of the signal [to 3forb,4forb, . . . ] which (despite being subdominant) increase the richness of the waveform.

With this, in the restricted PN approximation, the GW signal observed from the SolarSystem Barycenter (SSB) takes the following simple form:

h+(τ ;λsrc, ι) =

[1 + cos2 ι

2

]an cosφgw ,

h×(τ ;λsrc, ι) = [cos ι] an sinφgw . (2.17)

In general, all quantities may change over time, although in the case of short-lived, non-precessing binaries, the angle ι will be constant and therefore, the time dependence willcome from an = an(τ ;λsrc) and φgw = φgw(τ ;λsrc) in the form of a chirping signal, i.e. asinusoidal signal with amplitude and frequency increasing with time (see Chapter 3 forfurther details).

Looking at the leading order in the PN expansion for h22, e.g. in [4], we obtain that

an(τ ;λsrc) = −4νM

DLv2 , (2.18)

where M = m1 + m2 is the total mass of the system, ν = m1m2M2 is the symmetric mass

ratio, DL is the luminosity distance to the source and v = (πfM)1/3 is a characteristicvelocity of the two orbiting objects which also has the role of a PN expansion parameter.Here we keep an explicit negative sign coming from the second derivative of a complexexponential function.

Including all (`,m) modes

Let us now consider the most general case, where all the spherical harmonic components areincluded in the gravitational waveform. Formally, this calculation is equivalent to what wehave done previously to obtain Eq. (2.15) and indeed the first thing that we shall consideris the relation between the h`±m components given by (2.9). Moreover, we notice that any


of the (s = −2) spin-weighted spherical harmonics listed in Table 2.1 can be generallywritten as

−2Y`m(ι, β) = −2Y

`m(ι, 0)eimβ (2.19)

where −2Y`m(ι, 0) is always a real quantity2; therefore, making use of (2.6),

−2Y`−m(ι, β) = (−1)` −2Y

`m(π − ι, 0)e−imβ . (2.20)

With these results, we can expand the expression given in Eq. (2.2) and in particular, weshall write the contribution from a particular ` and ±m 6= 0 (both signs) values as

−2Y`m h`m + −2Y

`−m h`−m = [hR`m + ihI`m] −2Y`m(ι, 0)e−im(φorb−β) (2.21)

+(−1)2`[hR`m − ihI`m] −2Y`m(π − ι, 0)eim(φorb−β) ,

and taking into account that ` is an integer number and doing some basic algebra, onefinally obtains the following result:

(2.21) = S`m(ι)(hR`m cos[m(φorb − β)] + hI`m sin[m(φorb − β)]

)−i D`m(ι)

(hR`m sin[m(φorb − β)]− hI`m cos[m(φorb − β)]

), (2.22)

where the real functions S`m(ι) and D`m(ι) have been defined as(S`mD`m

)(ι) ≡ −2Y

`m(ι, 0)± −2Y`m(π − ι, 0) , (2.23)

with S standing for ‘sum’ and D for ‘difference’. Given the explicit expressions of the spin-weighted spherical harmonics in Tab. 2.1, it is straightforward to obtain the analyticalexpressions of these two functions too (see Table 2.2 for the expressions up to ` = 6).

Equation (2.22) is nothing but a generalization, for arbitrary (`,m) values, of the re-sult presented above [Eq. (2.15)] for the (2, 2) mode. All the (m-th) modes contribute as(co)sinusoidal functions of m

2 φgw, so all the dependency on the angle β [we recall thatit is an angle measured over the orbital plane from a fixed direction] can be absorbed inthe definition of the GW phase (2.16). Moreover, one can see directly from the definitionof S`m(ι) and D`m(ι) that the ‘+’ polarizations [sum over real parts of (2.22)] and ‘×’polarizations [sum over (minus) imaginary parts of (2.22)] will satisfy, in general [recallthe definition in Eq. (2.2)], the same symmetry property, under the transformation of theangle ι into its supplementary, as the S`m(ι) and D`m(ι) functions, i.e. :

h(+×)(τ ;λsrc, π − ι) =

(+−)h(+×)(τ ;λsrc, ι) . (2.24)

This general property, that has naturally appeared in our calculations, will be very im-portant from a data analysis point of view, on the one hand because it will reduce theparameter space to be explored by a factor two, going from a possible range of ι ∈ [0, π] [werecall that ι was a mod(π) angle by definition: angle between two directions] to the prac-tical range ι ∈ [0, π/2] and, on the other hand, because it may create some degeneraciesin the parameter space.

2Indeed, this can be formally demonstrated from the definition of the (s = −2) spin-weighted sphericalharmonics, Eq. (2.3).


The cases with m = 0 do not require to sum the positive and negative indexes as they arejust one single term. However, they can be formally written as Eq. (2.22) [with just oneterm in the l.h.s.] by defining

S`0(ι) = D`0(ι) ≡ −2Y`0(ι, 0) . (2.25)

Finally, the generic sum given at the very beginning of this discussion, Eq. (2.2), can berewritten as sums of sinusoidal functions of the different harmonics of the orbital phase;obtaining:

h+(τ ;λsrc, ι) ≡∞∑m=0

[u+,m cos

(m2 φgw

)+ w+,m sin

(m2 φgw

)],

h×(τ ;λsrc, ι) ≡∞∑m=0

[u×,m cos

(m2 φgw

)+ w×,m sin

(m2 φgw

)], (2.26)

where the factors (u,w)(+,×),m can be obtained from the spin-weighted spherical harmon-ics, h`m, explicitly given in Ref. [4] and the ‘sum’ and ‘difference’ angular functions writtenin Tab. 2.2. Let us recall that all the dependency on β has been absorbed in the definitionof the GW phase φgw(τ ;λsrc), see Eq. (2.16), and the amplitude terms can be written as

u+,m(τ ;λsrc, ι) ≡∑

` S`m(ι) hR`m(τ ;λsrc) ,

u×,m(τ ;λsrc, ι) ≡ −∑`D`m(ι) hI`m(τ ;λsrc) ,

w+,m(τ ;λsrc, ι) ≡∑

` S`m(ι) hI`m(τ ;λsrc) ,

w×,m(τ ;λsrc, ι) ≡∑

`D`m(ι) hR`m(τ ;λsrc) ,

(2.27)

where have used a shortened notation for the sum symbol,∑

` ≡∑∞

`=max(2,m). Moreover,note that in the previous expressions, the double sum over ` and m has been rewrittenfrom the original

∑∞`=2

∑`m=0 in Eq. (2.2) after pairing the ±m terms off, to the equivalent∑∞

m=0

∑∞`=max(2,m).

When one is working with the PN formalism, each spherical harmonic component h`m iswritten as a PN expansion series [4] and therefore, each of the quantities that appear inEqs. (2.27) is indeed a sum over ` and over the PN order, n.

After a careful study of the quantities in Eq. (2.27), either within the effective-one-body(EOB) formalism or looking directly at the PN expansion terms, one finds out that thereare some common terms that can be factored out. On the one hand, we shall factorthe ‘Newtonian’ amplitude out, an [see definition in Eq. (2.18)], since it contains all theinformation about the order of magnitude and units of the GW amplitude; on the otherhand, there are several factors common to all the u(+,×),m and w(+,×),m functions of aparticular m-th harmonic [7] and we shall see later that it is indeed very convenient toalso factor them out, mainly because these factors may be zero at some point and havingthem explicitly factorized will help avoiding possible numerical divergences. In particular,we shall define a new u(+,×),m and w(+,×),m functions as follows:(

u(τ ;λsrc, ι)

w(τ ;λsrc, ι)

)(+,×),m

≡ an(τ ;λsrc)Υm(λsrc, ι)

(u(τ ;λsrc, ι)

w(τ ;λsrc, ι)

)(+,×),m

, (2.28)


where the functions Υm (for m ≤ 8) are defined as

Υm=0(λsrc, ι) ≡ 1 Υm=1(λsrc, ι) ≡ δmM sin ι

Υm=2(λsrc, ι) ≡ 1 Υm=3(λsrc, ι) ≡ δmM sin ι

Υm=4(λsrc, ι) ≡ sin2 ι Υm=5(λsrc, ι) ≡ δmM sin3 ι

Υm=6(λsrc, ι) ≡ sin4 ι Υm=7(λsrc, ι) ≡ δmM sin5 ι

Υm=8(λsrc, ι) ≡ sin6 ι

. (2.29)

[We recall that M = m1 + m2 is the total mass of the binary system, and δm = m2 −m1 is the mass difference.] Besides this, we have noticed that properly (trigonometric)expanding all the D`m(ι) functions, they turn out to be proportional to ‘cos ι’ and so all‘×’ polarization terms are, i.e. (u,w)×,m ∝ cos ι for all m. We do not write it explicitlysince we shall not use this property, although we found useful to let the reader know aboutit.

We shall see below that the notation used in Eqs. (2.26) and (2.28) is very convenient whenexpressing the measured strain in a GW detector as a single cosine function [see Eq. (2.55)below] and therefore, this formal expression for h+ and h× as an harmonics expansion issometimes taken as the starting point for many data analysis studies [7–11]. Since fromnow on, there shall not be any more explicit references to the spherical harmonics, andin order to not confuse the m-th harmonic with the any ‘mass’ parameter, a change ofnotation is usually adopted at this point, replacing m by j. Thus, from this moment on,we shall refer any of the terms of the sum in Eq. (2.26) as the j-th harmonic of the GWsignal:

j ≡ m . (2.30)

2.2 Detected gravitational wave strain

The GW signal described in the previous section as the superposition of two orthogonalpolarizations, will travel, as it is predicted by the theory of general relativity, almostunaltered [given the almost null interaction of gravitational radiation with matter] at thespeed of light from the violent source that generated it to the observer at Earth. Thus, ina Cartesian coordinate system tied to the wave’s propagation, x′, y′, z′, where the GWtravels in the +z′ direction and the perturbations are contained within the x′ − y′ plane,with the x′ direction being the main axis of the ‘+’ polarization, the GW tensor in simplygiven by (see Sec. 1.1)

H ′ = h+ [x′ ⊗ x′ − y′ ⊗ y′] + h× [x′ ⊗ y′ + y′ ⊗ x′] , (2.31)

where x′ / y′ represents a unit vector parallel to the x′ / y′ axis, ‘⊗’ denotes the tensorialproduct and h(+,×) ≡ h(+,×)(τ ;λsrc, ι) are the orthogonal polarizations given in Eq. (2.26)as a function of the orientation angle, ι; the GW phase, φgw(τ ;λsrc); and the differentspherical harmonics components, h`m(τ ;λsrc).

As a metric perturbation, the effects of a GW consist on stretching and compressingthe space in a perpendicular plane to the propagation direction. These are, precisely, theeffects that one will measure with an interferometric detector and in particular, the length

2.2. Detected gravitational wave strain 33

` = 2

S22 = 14

√5π (1 + cos2 ι) D22 = 1

2

√5π cos ι

S21 = 12

√5π sin ι D21 = 1

4

√5π sin 2ι

` = 3

S33 = − 116

√212π (5 sin ι+ sin 3ι) D33 = − 1

4

√212π sin 2ι

S32 = 12

√7π cos 2ι D32 = 1

16

√7π (5 cos ι+ 3 cos 3ι)

S31 = − 116

√352π (sin ι− 3 sin 3ι) D31 = 1

4

√352π sin 2ι

` = 4

S44 = 316

√7π (3 + cos ι) sin2 ι D44 = 3

4


S43 = 316

√72π (sin ι− 3 sin 3ι) D43 = − 3

2

√72π cos3 ι sin ι

S42 = 332√π

(5 + 4 cos 2ι+ 7 cos 4ι) D42 = 316√π

(cos ι+ 7 cos 3ι)

S41 = 316√2π

(3 sin ι+ 7 sin 3ι) D41 = 316√2π

(−2 sin 2ι+ 7 sin 4ι)

` = 5

S55 = − 116

√1652π (3 + cos 2ι) sin3 ι D55 = − 1

4


S54 = 14

√33π (1 + 2 cos 2ι) sin2 ι D54 = 1

32

√33π (19 cos ι+ 5 cos 3ι) sin2 ι

S53 = − 1128

√332π (14 sin ι+ 13 sin 3ι+ 15 sin 5ι) D53 = 1

32

√332π (2 sin 2ι− 9 sin 4ι)

S52 = 18

√11π (cos 2ι+ 3 cos 4ι) D52 = 1

64

√11π (14 cos ι+ 3(cos 3ι+ 5 cos 5ι))

S51 = 1128

√77π (−2 sin ι− 3 sin 3ι+ 15 sin 5ι) D51 = 1

32

√77π (2 sin 2ι+ 3 sin 4ι)

` = 6

S66 = 3128

√715π (3 + cos 2ι) sin4 ι D66 = 3

32


S65 = − 164

√2145π (3 + 5 cos 2ι) sin3 ι D65 = − 1

128

√2145π (13 cos ι+ 3 cos 3ι) sin3 ι

S64 = 1256

√1952π (67 + 92 cos 2ι+ 33 cos 4ι) sin2 ι D64 = 1

32

√1952π (13 cos ι+ 11 cos 3ι) sin2 ι

S63 = 3512

√13π

(10 sin ι+ 3 sin 3ι− 55 sin 5ι) D63 = − 31024

√13π

(51 sin 2ι+ 20 sin 4ι+ 55 sin 6ι)

S62 = 12048

√13π

(289 cos 2ι+ 30(7 + cos 4ι) + 495 cos 6ι) D62 = 1512

√13π

(10 cos ι+ 81 cos 3ι+ 165 cos 5ι)

S61 = 1256

√652π

(10 sin ι+ 27 sin 3ι+ 33 sin 5ι) D61 = 1512

√652π

(−17 sin 2ι− 12 sin 4ι+ 99 sin 6ι)

Table 2.2: ‘Sum’ [S`m ≡ S`m(ι)] and ‘difference’ [D`m ≡ D`m(ι)] functions defined in Eq. (2.23) andcomputed from the explicit expressions of the (s = −2) spin-weighted spherical harmonics listed in Tab. 2.1


difference between the two arms of the interferometer. The aim of this section is to translatethe general tensor expression for a gravitational signal written in the wave Cartesiancoordinate system, Eq. (2.31), into the measured differences between the wave-inducedrelative length changes of the two interferometer arms, which is an scalar3 quantity. Theresponse of a laser interferometric detector to a weak, plane, GW on the long wavelengthapproximation (LWA) (i.e. when the size of the detector is much smaller than the reducedwavelength λ/(2π) of the wave) is well known [12] and can be computed as4

h(t) ≡ ∆Larms(t)

L=

1

2nT1 ·H · n1 −

1

2nT2 ·H · n2 . (2.32)

Here, n1 and n2 denote the unit vectors parallel to the two detector’s arms (the orderof arms is defined such that the vector n1 × n2 points outwards from the surface of theEarth), H is the 3-dimensional matrix of the spatial metric perturbation produced by thewave, and a dot stands for the standard scalar product in the 3-dimensional Cartesianspace. Notice that, since Eq. (2.32) is an scalar expression, it can be computed in anycoordinates system.

The long wavelength approximation is completely valid for current and advanced ground-based interferometers, as they are [at most] 4 km long and their frequency window’s high-end, f < 2000 Hz, translates into a reduced wavelength of λ

2π > 24 km. The future 3rdgeneration interferometers, such as ET, are planned to be Let = 10 km long, and thereforeone might consider going beyond the LWA for very high frequency sources. LISA, on theother hand, is going to have an arm-length of Llisa = 5 × 106 km and it is expectedto observe signals up to f < 0.1 Hz, which translates into λ

2π > 0.45 × 106 km. Thus,LISA is going to have a non-negligible fraction of its frequency window where one musttake into account the GW phase variation as the laser light of the interferometer travelsbetween the two mirrors. The resulting response of a laser interferometric detector beyondthe LWA is also well-known [13], namely the time delay interferometry (TDI), and involvethe evaluation of the GW waveform at different instants. We shall not go into any moredetails about TDI in this thesis, since the LWA has been assumed to be valid in all thestudies we have performed, which is strictly true for all the ground-based detectors studies(Chapter 8) and a very good approximation for all the LISA studies that we have carriedout (Chapters 4-7), since most of the sources considered lie at the very low part of theLISA’s frequency window: f ∼ 10−4 Hz, which means λ

2π ∼ 500× 106 km Llisa.

Before proceeding with the derivations, let us properly introduce the three Cartesian co-ordinate systems that we shall consider: a wave coordinate system x′, y′, z′ in where theGW travels in the +z′ direction; the “barred” coordinates x, y, z tied to the detector;and the “unbarred” coordinates x, y, z fixed to the ecliptic plane and therefore, untiedto the detector’s motion. In particular,

• x′, y′, z′ is the wave Cartesian coordinate system, where the GW travels in the +z′

direction and the node direction of the ’+’ polarization is parallel to the x′ axis. Inthis frame, the 3-dimensional matrix of the spatial metric perturbation, H ′, is simply

3By scalar, we mean that the result is invariant under Cartesian coordinate transformations.4We recall that the relative length change produced by a GW is ∆L

L= h

2. Now, considering a two-arms

interferometer and given the quadrupole nature of gravitational radiation, when one arm is stretched ina certain direction, it is compressed in the orthogonal one; i.e. Lx = L + ∆L and Ly = L − ∆L, hence∆Larms

L≡ Lx−Ly

L= 2 ∆L

L= h.


Figure 2.1: Two different perspectives from a 3-dimensional diagram representing the most useful co-ordinates systems and angles to describe the generation (source coordinate system), propagation (waveCartesian coordinate system, x′, y′, z′) and detection (frame x, y, z attached to the GW detector) ofGW signals. [Diagrams: own production]


Eq. (2.31) and, in matrix form

H ′ =

h+ h× 0h× −h+ 00 0 0

. (2.33)

• x, y, z is a reference frame attached to the GW detector, having the detector’s planecontained inside the x − y plane and the z axis pointing toward zenith. FollowingCutler’s [3] criterion, the two detector’s arms, assumed to form an arbitrary angle ζ;are ”symmetrically placed” inside the x− y axes (see Fig. 2.1), i.e. both, detector’sarms and x− y axes, share the angle bisector.

• Finally, x, y, z is an ecliptic Cartesian coordinates system centered at the SSB andwith the x− y plane tied to the ecliptic (i.e. the plane of the Earth’s motion aroundthe Sun). The x axis points toward an arbitrary but fixed direction in the sky andz is chosen in order to have the Earth’s angular momentum around the Sun to beparallel to the +z direction.

2.2.1 Gravitational wave strain in the detector’s coordinates frame

In Fig. 2.1 we represent x′, y′, z′ and x, y, z frames and the relevant angles that takepart in the coordinates transformation. In particular, the 3-dimensional orthogonal matrixof transformation, M , (from the wave Cartesian coordinates x′, y′, z′ to the Cartesiancoordinates x, y, z in the detector’s proper frame) can be expressed as a compositionof Euler rotations [14] within the Z-Y-Z form (the most common convention in GW dataanalysis). Moreover, since z′ is parallel to the −N vector, it turns out that the θ, φEuler angles of transformation from x, y, z to x′, y′, z′, coincide (modulo π) with thepolar angles, θn, φn, of the unit vector N in the detector’s reference frame. The thirdEuler angle, ψ, is the polarization angle and is the angle (see Ref. [2]) from the principaldirection of the ’+’ polarization: ±N× L; to the direction of constant azimuth (φn) overthe x′ − y′ plane: ±N× (N× z), where z is the unit vector parallel to the z axis. Since ψis defined as an angle between two directions, it will be modulo π and we arbitrarily set itto be ψ ∈ [0, π], i.e. its sine is positive defined. Making use of the properties of the scalarand vectorial products between vectors in a 3-dimensional Cartesian space, we obtain thefollowing scalar expressions to determine the polarization angle:

cosψ = N · (L× z) , sinψ = |L · z− (L · N)(z · N)| . (2.34)

With this, the orthogonal matrix that transforms x′, y′, z′ 7→ x, y, z, M , can be ex-pressed as the product of three Euler rotation matrices5 [14]:

M = Rz(−φn) ·Ry(−θn + π) ·Rz(ψ)

=

cos(−φn) − sin(−φn) 0

sin(−φn) cos(−φn) 00 0 1

· cos(−θn + π) 0 sin(−θn + π)

0 1 0

− sin(−θn + π) 0 cos(−θn + π)

· cosψ − sinψ 0

sinψ cosψ 00 0 1

=

− cosφn cos θn cosψ + sinφn sinψ sinφn cosψ + cosφn cos θn sinψ cosφn sin θnsinφn cos θn cosψ + cosφn sinψ cosφn cosψ − sinφn cos θn sinψ − sinφn sin θn

− sin θn cosψ sin θn sinψ − cos θn

.

(2.35)

5Note that since N = −z′, the angles appearing in the rotation matrices are not the trivial ones.


In the detector’s Cartesian coordinate system x, y, z, the 3-dimensional matrix of thespatial perturbation produced by the wave can be expressed as a coordinates transforma-tion of the GW signal in the wave’s frame, H ′, written in Eq. (2.33):

H = M ·H ′ ·MT , (2.36)

and the unit vectors parallel to the arms, n1 and n2, can be straightforwardly written as

n1 =

cos(π4 −ζ2)

sin(π4 −ζ2)

0

n2 =

sin(π4 −ζ2)

cos(π4 −ζ2)

0

. (2.37)

With this, we have all the quantities needed to compute the measured scalar GW strain inthe interferometer, h(t) [see Eq. (2.32)], expressed in the very same coordinates, x, y, z.Putting Eqs. (2.32)-(2.37) together, we obtain the following final expression for h(t)

h(t;λsrc, N, ι, ψ, ζ) = F+(t; N, ψ, ζ) h+(τ(t);λsrc, ι) + F×(t; N, ψ, ζ) h×(τ(t);λsrc, ι) ,(2.38)

where we recall that h(+,×) are the orthogonal polarizations given in Eq. (2.26), and

F+(t; N, ψ, ζ) = sin ζ[

12(1 + cos2 θn) cos(2φn) cos(2ψ)− cos θn sin(2φn) sin(2ψ)

]F×(t; N, ψ, ζ) = − sin ζ

[12(1 + cos2 θn) cos(2φn) sin(2ψ) + cos θn sin(2φn) cos(2ψ)

](2.39)

are the so-called antenna beam patterns and they depend on the sky position of the GWsource [θn(t), φn(t)], the polarization angle [ψ(t)] and the angle between the two interfer-ometer’s arms [ζ]. Notice that having the interferometer’s arms with an angle smaller than90, i.e. ζ > 0, is equivalent as having a 90 arms with an effective length equal to L sin ζ.In other articles, for instance Ref. [2, 3], the explicit minus sign that we are finding in frontof F× is absorbed in the definition of h×, Eqs. (2.17) or (2.26), moving the explicit minussign from F× to h×; in any case the final expression for measured strain is identical.

Making use of the straightforward properties:

(i) F+(ψ′ = π4 + ψ) = F×(ψ) and (ii) F+,×(π2 + ψ) = −F+,×(ψ) ;

we represent in Figs. 2.2 and 2.3 the plots of the antenna beam pattern functions (inabsolute value) over the sky for a single 90 arms’ detector and considering several valuesof the angle ψ. Notice that, in each hemisphere, there are two [approximated] azimuthaland orthogonal directions where the detector is most sensitive and two directions [rotatedπ/4 with respect to the first ones] where the sensitivity is minimum. Indeed, these anglescan be obtained analytically; in particular for the |F×| function, the maxima for the zenith[θ = 0] and nadir [θ = π] are found at φθ=0 = −ψ + π

4 + nπ2 and φθ=π = ψ + π4 + nπ2 ,

respectively, n being an integer number. [The minima are rotated π/4 with respect to thesedirections.] Notice that the angle difference between the maxima directions at the nadirand the zenith is (φθ=π − φθ=0) = modπ

2(2ψ); as it can be seen in Figs. 2.2 and 2.3.


Figure 2.2: Plot of a single detector’s antenna beam pattern function (in absolute value) over the skyfrom three different view angles. In particular, we are plotting the following two equivalent functions ofthe spherical angles: |F×(θ, φ, ψ = π

8, ζ = π

2)| = |F+(θ, φ, ψ′ = 3π

8, ζ = π

2)|. [Plot: own production]

ψ = 0 ψ = π12 ψ = π

6 ψ = π4 ψ = π

3 ψ = 5π12 ψ = π

2 or 0

ψ′ = π4 ψ′ = π

3 ψ′ = 5π12 ψ′ = π

2 or 0 ψ′ = π12 ψ′ = π

6 ψ′ = π4

Figure 2.3: Sequence of graphical representations of the antenna beam patterns functions (in absolutevalue) for a single 90 arms’ detector, |F×(ψ)| = |F+(ψ′ = π

4+ ψ)| [see Eq. (2.39) and Fig. 2.2], plotted

over the sky and for different values of the polarization angle ψ. Since, F+,×(π2

+ ψ) = F+,×(ψ) we justrepresent the interval ψ ∈ [0, π

2]. [Plots: own production]


2.2.2 Change of apparent sky location and polarization due to detector’smotion

Equation (2.38) represents the general expression of the measured GW strain in an inter-ferometric detector in the long wavelength approximation (LWA). This result is writtenin terms of angles measured at the detector’s Cartesian coordinate system, which followsthe detector’s movement around the Sun; thus, the “barred” angles θn, φn, ψ will changeover the observation time when this is long enough (ζ is constant). In particular, ground-based detectors are expected to only observe the late inspiral and merger parts of theCBC evolution, which represents, at most, the last ∼ 200 cycles of the GW signal. Sincethe lowest frequency that ground-based detectors can observe is ≈ 20 Hz, this means thatthe longest6 CBC signal observed in Earth will last 10 sec, over which the Earth’s motioncan be completely neglected. Thus, for ground-based observations of CBC signals, the an-tenna beam patterns can be considered as constant functions in time, that only dependon the particular interferometer and date time of the observation at hand. For long-livedsources, such as the stochastic background and the gravitational emission of rapidly spin-ning pulsars, the antenna beam patterns become a time dependent functions that provideinformation about the sky location of the source. We refer the reader to Ref. [12] for adetailed explanation about how to include the Earth’s motion in long-lived GW signalsfor ground-based detectors.

The LISA case, however, is a completely different story; mainly for two reasons. First,because LISA will observe a much lower frequency window, [10−5, 0.1] Hz, but keeping thesame intrinsic orbital period of 1 yr, since LISA is following the Earth in its motion aroundthe Sun; this means that the same 200 GW cycles would last up to ≈ 8 months. Second,because LISA will observe CBC signals coming from supermassive black-hole systems withsignal-to-noise ratios of several hundreds, which means that it can observe signals that arestill far from the merger (inspiral-only signals), which may span several thousands of cycles[this is the case, for instance, of the inspiral signals that we shall study in Chapters 4 and5 and the galactic binaries of Chapter 6]. Thus, LISA observations of CBC may last thewhole mission life-time and therefore we must consider the detector’s motion around theSun. This problem has been studied in detail by Cutler [3] and here we just quote theirmain results [making a careful conversion between their and our notation] that explicitlyexpress the time dependence of the “barred” angles θn, φn, ψ in terms of the constantangles measured from the SSB.

cos θn(t; N,λlisa) =1

2cos θn −

√3

2sin θn cos[φlisa(t;λlisa)− φn] , and (2.40)

φn(t; N,λlisa) = α0 +2πt

Tlisa+ arctan

(√3 cos θn + sin θn cos[φlisa(t;λlisa)− φn]

2 sin θn sin[φlisa(t;λlisa)− φn]

),

(2.41)where Tlisa = 1 yr represents LISA’s orbital period, φlisa(t;λlisa) = φlisa, 0 + 2πt

Tlisais the

angular position of LISA over its orbital plane, and φlisa, 0 and α0 are just constantsspecifying, respectively, the detector’s location in the orbital plane and the orientation ofthe arms at time t = 0.

6These are very conservative numbers. In reality typical CBC observations in ground-based GW detec-tors will last fractions of a second.


Finally, the polarization angle, ψ(t; N, L,λlisa), can be computed from its scalar definitionin Eq. (2.34), taking into account that the vectors that appear in there can be expressedin the SSB reference frame as,

N =

sin θn cosφnsin θn sinφn

cos θn

; L =

sin θl cosφlsin θl sinφl

cos θl

; z =

−√

32 cosφlisa(t;λlisa)

−√

32 sinφlisa(t;λlisa)

12

,

(2.42)where one has used the fact that the normal to the detector plane, z, is at a constant 60

angle to z, and z precesses around z at a constant rate. The explicit expressions for ‘sinψ’and ‘cosψ’ can be found in Eqs. (3.19)-(3.22) of Ref. [3].

2.2.3 Doppler shift due to relativistic effects and relative motion source– detector

All the expressions for the emitted GW signal derived in Sec. 2.1 are given in terms of theproper timing of the source’s barycenter reference frame, τ ≡ τe. Thus, we must considerthe relativistic effects that link this proper time of emission, τe, to the proper time ofarrival, τa, at the detector’s reference frame. In order to do so, we shall make use of theknowledge acquired in the 1980s about ‘timing’ relations when studying the experimentaldata from the binary pulsar PSR 1913+16; in particular we shall use as reference thearticles [15, 16].

Assuming a system of harmonic coordinates at rest with respect to the SSB, the ‘timingformula’ linking the proper time of emission, τe, to the coordinate time of arrival at thedetector, ta, can be written as [15, 16]

D τe = ta −∆S +rifo · N

c+ ∆E, −

DL

c. (2.43)

Here, ∆S denotes the general-relativistic Shapiro time-delay due to the propagation of theGW signal in a curved space-time caused by any gravitational field in the path betweenthe source and the detector (e.g. the Sun); rifo ·N/c is the (coordinate) Roemer time-delay,i.e. the travel time along rifo ≡ rifo(t), from the SSB to the detector; and ∆E, is theso-called Einstein time-delay in the solar system, which represents a combined effect ofgravitational redshift and time dilation due to motions of the Earth and other bodies [16].The last term in the expression is just a constant distance between the source barycenterand the SSB at a given time, since all the time dependency of the distance separationbetween the two barycenters is included in the Doppler factor, D, [15]

D ≡ 1 + vsrc·Nc√

1− v2srcc2

= γ(

1 + vsrc·Nc

)≡ (1 + z) , (2.44)

where z is simply the redshift.

The effects of both Shapiro and Einstein time-delays are generally much smaller than theNewtonian Roemer-delay, and therefore the ‘timing relation’ used in GW data analysis ofsignals from the High- and Low-Frequency bands is simply

D τe = ta +rifo · N

c+ const. (2.45)


This relation has basically two effects on the measured GW signal:

• On one hand, we are measuring redshifted time and frequency quantities; i.e. theobserved measurements will be related to the physical ones (as measured in thesource reference frame) as follows:

tobs = (1 + z) tphys ; fobs =fphys1 + z

,

where we recall that (1 + z) ≡ D. Given the fact that all the CBC dynamics isformulated in terms of the adimensional time and frequency variables, t/M and fM ,respectively (see Chapter 3); this redshift effect is normally absorbed by definingobserved redshifted masses, which are related to the physical ones simply as

mobs = (1 + z)mphys , (2.46)

‘m’ being any dimensionful mass, i.e. total mass M , mass difference δm, individualmasses m1,m2, reduced mass µ, or chirp mass M. Of course, the dimensionlessmass quantities, such as the mass ratio q or the symmetric mass ratio ν, are notredshifted.

• On the other hand, the Roemer time-delay will produce a Doppler shift in the ob-served GW phase, that generally can be represented as

φgw(τ) = φgw(τ(ta)) = φgw

(ta+rifo·N/c

D

). (2.47)

– For low frequency signals (such as LISA band), the GW phase does not changevery much during the travel time along rifo, which makes possible to seriesexpand φgw and represent the Doppler-shift effect as an additive phase:

φgw(τ) = φgw

(taD

)+

rifo · Nc

φgw

(taD

)+O

[rifoc

]2 ' φgw + ϕd , (2.48)

where we have neglected second-order Doppler corrections since they are of or-der (vifo/c)|ϕd(t)| . 3×10−4(f/10−3) rad [3]. The dot denotes a time-derivativein the detector’s reference frame (i.e. with respect to t) and the Doppler phaseϕd is defined as

ϕd(t) =2πF (t)

crifo · N , (2.49)

where F (t) ≡ φgw(t)2π is the (redshifted) GW frequency.

Making use of the explicit expression for the position of the LISA detector as afunction of time [3],

rlisa(t) = Rlisa

cosφlisa(t)sinφlisa(t)

0

, (2.50)

where Rlisa = 1 AU and φlisa(t) is the angular position of LISA over its orbitalplane [see paragraph right after Eq. (2.41)]; one can get an explicit expressionfor the Doppler phase for LISA [3]:

ϕd, lisa(t) =2πF (t)

cRlisa sin θn cos(φlisa(t)− φn) . (2.51)


– For higher frequency signals (e.g. ground-based detectors band), one can not usethe expansion in (2.48), although similar expressions can be obtained when theexplicit time dependency of the phase is known; see, for instance, the derivationin Appendix A of [12] for GW signals from spinning NSs.

2.3 Putting all pieces together: measured GW strain as asum of cosine functions

In this chapter we have gone from the raw output of a numerical simulation, or a PN

calculation (the Regge-Wheeler and Zerilli functions, Ψ(o)`m and Ψ

(e)`m), to the very ‘end

product’, this is the time series GW strain measured in the detector, h(t). The process canbe summarized in two steps; first, an expansion over the spin-weighted spherical harmonicsthat provide the GW components with the corresponding angular dependency, here iswhere we have defined the (u, w)(+,×),m functions that expand h+ and h× in terms ofharmonics of the GW phase; and second, a projection of the GW spatial tensor intothe detector’s frame, which has defined the antenna beam patterns and has added thedependency on the sky location and the detector’s motion to the measured signal, h(t).

At this point, it is time to place everything in order and to write explicitly the generalexpression for a time-domain GW signal emitted by a CBC and measured by an interfer-ometric GW detector. This is, indeed, the purpose of this section.

Let us start considering the restricted PN approximation, where h+ and h× are given byEq. (2.17); the former containing just a ‘cosφgw’ and the latter with a sine of the samephase. Making use of the general expression for the measured differences between the wave-induced relative length of the two interferometer arms, Eq. (2.38), and basic trigonometricrelations, one can write h(t;λsrc, N, ι, ψ, ζ) ≡ h(t) as

h(t) =

[1 + cos2 ι

2

]F+ an cos(φgw + ϕd) + [cos ι] F× an sin(φgw + ϕd)

≡ Cp an cos(φgw + ϕp + ϕd) , (2.52)

where

Cp ≡1

2

√(1 + cos2 ι)F 2

+ + 4 cos2 ιF 2× , (2.53)

ϕp ≡ arctan

[ −2F× cos ι

F+(1 + cos2 ι)

]. (2.54)

The subscript ‘p’ stands for polarization and ϕp is called polarization phase. If we were con-sidering observations from multiple detectors, the amplitude and phase differences betweenthe various detected strains would involve different Cp and ϕp values through the antennabeam patterns of each detector, and therefore we could infer the sky location θn, φn andpolarization angle ψ of the source. Also, if we are observing a GW signal for a long timeperiod (e.g. CBCs with LISA), the time dependence of these two values, Cp and ϕp, canbe used to estimate the position of the source. Finally, if one considers short-lived signalsdetected by a single interferometer, Cp and ϕp will be numerical constants and absorbed,respectively, into an effective distance, Deff ≡ DL/Cp [where DL is the luminosity distance]and the arbitrary initial phase of the signal, φ0 ≡ φgw(t = 0).

2.4. Residual gauge-freedom in the measured GW strain 43

A similar derivation can be done for the general case, where h(t) is written as the superpo-sition of different harmonics of the orbital phase, see Eq. (2.26). Let us recall that usually,in the data analysis articles, the different harmonics are denoted with the letter j insteadof the m used in Sec. 2.1 [see Eq. (2.30)]. In this case, one shall find a different factor Cp,jand polarization phase, ϕp,j , for each harmonic:

h(t) = an

∞∑j=0

Υj

[(F+ u+,j + F× u×,j) cos( j2φgw + ϕd) + (F+ w+,j + F× w×,j) sin( j2φgw + ϕd)

]≡ an

∞∑j=0

Υj Cp,j cos( j2φgw + ϕp,j + ϕd) , (2.55)

where

Cp,j ≡√

(F+ u+,j + F× u×,j)2 + (F+ w+,j + F× w×,j)2 , (2.56)

ϕp,j ≡ arctan

[−(F+ w+,j + F× w×,j)

(F+ u+,j + F× u×,j)

]. (2.57)

Here is where the benefits of factorizing the common terms out of (u,w)(+,×),j becomerelevant since (i) an appears in the explicit expression of h(t) carrying the units and orderof magnitude of the amplitude and (ii) the Υj functions cancel out in the definition of thepolarization phase, which shall prevent potential numerical divergencies in cases where δmor ι are close to zero [see Eqs. (2.29)].

Let us notice that, both in Eqs. (2.54) and (2.57), the sign of the fraction inside the ‘arctan’function is explicitly written in the numerator in order to remove the mod(π) degeneracyof the polarization phase, by making explicit that the sign of ‘sinϕp,j ’ is the sign of thenumerator, and the same for ‘cosϕp,j ’ and the denominator. Sometimes, this two-argument‘arctan’ function is called “atan2”.

2.4 Residual gauge-freedom in the measured GW strain

The measured GW strain depends on a lot of parameters. Some of them, such as λifo, ζare fixed by the detector’s geometry and motion; others, such as λsrc depend only onthe physical properties of the source; and finally, N, φ0, ι, ψ depend on the relativeorientation of the detector with respect to angular momentum of the compact binary. Itis, precisely, this last set of parameters, the ones that explicitly appear in the relationbetween the measured h(t) and the emitted spherical harmonics components h`m, andthey contain some residual gauge-freedom (symmetries) that will make the measured GWstrain [approximately] invariant under certain transformations of these parameters.

Studying and restricting this residual gauge-freedom in h(t) is essential in order to removeequivalences or degeneracies between different regions of the parameter space. We shallsee that some of the symmetries are exact, and therefore we shall be talking about equiv-alence between different regions; whereas some of them will be valid just as a limit caseor approximately, in this case, we shall talk about degeneracies in the parameter space.The benefits of detecting such symmetries are two-fold; on the one hand, by finding twoequivalent regions of the parameter space, one shall be able to restrict the range of somethe parameters and therefore, to reduce the search volume of parameter space; on the


other hand, knowing before-hand about the existence of degeneracies in the parameterspaces will help understanding the results out from a blind search, as well as improvingthe search method itself.

We shall analyze the symmetries of each of the functions that take part into the calculationof h(t), making always clear whether they are valid in general or simply under certain cir-cumstances. After that, we shall combine these symmetries in order to obtain the differentinvariant transformations.

• Let us start by the general and formally proven [see Eq. (2.24)] symmetry relationthat satisfy h+ and h× with respect to the orientation angle, ι:

(1) ι→ π − ι , h(+×) → ±h(+

×)

• Another symmetry related to the traveling GW signal that still does not involve theprojection of the GW tensor in terms of the antenna beam patterns, is related withthe arbitrary initial GW phase, φ0,

(2) φ0 → φ0 + π , h(+×) → −h(+

×)

However, this property (in the time-domain) is only valid for the j = 2 dominantharmonic, which is the only one not involving harmonics of φgw. Since the nextdominant harmonics [4] are the j = 1, 3 and j = 4, with amplitudes smaller thanthe j = 2 one by factors ∝ v δmM and ∝ v2, respectively [v 1 being a PN expansionparameter]; we can consider (2) as a good approximation, although not strictly true.

• Finally, the rest of symmetry relations come from the antenna beam patterns. Inparticular, we shall write them schematically as two summands and then study howdoes their sign change for different transformations of the angles θn, φn, ψ,

F+ = sin ζ[

12(1 + cos2 θn) cos(2φn) cos(2ψ)− cos θn sin(2φn) sin(2ψ)

]≡ a + b ,

F× = − sin ζ[

12(1 + cos2 θn) cos(2φn) sin(2ψ) + cos θn sin(2φn) cos(2ψ)

]≡ c + d .

Now, we consider all possible (but not repeated) angle transformations that changethe signs in front of a , b , c , d, but do not swap between these terms. Theyare summarized in the following table:

a b c d

original + + + +

(3) ψ → π2 − ψ − + + −

(4) ψ → π2 + ψ − − − −

(5) θn → π − θn + − + −(6) φn → π

2 − φn − + − +

(7) φn → π2 + φn − − − −

By combining (3)+(4) we would get that ψ → π − ψ gives [+ − −+] and similarresults could be obtained with (6)+(7) and (7)+(7); however we do not considerthem in the previous table since they can be obtained from a combination of othersalready included. Indeed, given the dependency of F+,× always with 2φn, we have

2.4. Residual gauge-freedom in the measured GW strain 45

that the amplitudes are invariant under φn → π + φn; this is (7)+(7). The samething happens with the polarization angle, ψ.

The three angles that we are considering are measured in the time-varying “barred”reference system, which means that the symmetry relations may not be valid whenone considers the constant “unbarred” angles, tied to the ecliptic reference framex, y, z. Moreover, one must take into account that the angles characterizing thesky location of the source, N, also affect the signal’s Doppler effect, which meansthere may be angle transformations that leave the amplitudes invariant, but changethe phase and frequency of the observed signal. In particular,

– (3) and (4) will be valid when the signal (or most of the SNR) is observedfor a short time [e.g. CBC observations with ground-based detectors (or signalsincluding the merger with LISA)]; or the normal to the detector’s plane, z, is onaverage, during Tobs, parallel to the ecliptic’s normal direction, z; which is thecase of LISA. In the former, the symmetry will be valid for the instantaneousψ, whereas in the latter it will be for the invariant ψ.

– The dependency of the Doppler phase due to (5) is null if the perpendicularcomponent to the ecliptic of the detector’s velocity is zero or can be neglected.This is the case for LISA and a very good approximation for ground-baseddetectors, where the perpendicular component can only come from the dailyspinning motion of the Earth; which, at most, can produce relative frequencychanges of ∆f

f ∼ 10−7. Thus, (5) will be valid in almost all typical cases con-sidered in GW data analysis.

– (6) and (7), on the other hand, involve transformations on φn that do affectsignificatively the Doppler phase. This means that although the amplitudes mayremain invariant, the observed phase will change unless it is compensated byalso changing some of the source physical parameters λsrc that can modify theGW phase, φgw. Thus, despite being transformations that may add degeneraciesto the parameter space (as we shall see in practice in the following chapters),we won’t consider them here as residual gauge-freedom.

Finally, let us note that there are also transformations that allow one to swap F+

for F×; in particular the ones involving ψ → π4 ± ψ ; ψ → 3π

4 ± ψ. Also, bydoing φ0 → π

2 ± φ0 or φ0 → 3π2 ± φ0, one can swap the ‘sinφgw’s for ‘cosφgw’s.

However, we are not considering them because we found not possible to then havean equivalent transformation in ι to also swap u+ for w×; and w+ for u×; i.e. S`m(ι)for D`m(ι) [here, the pairs are chosen so that they share hR`m or hI`m, see Eq. (2.27)].

With this, we find that symmetries (1)-(5), involving transformations on ι, φ0, ψ, θn, arevalid in good approximation. We discard the transformations involving φn, (6) and (7),because despite being valid for the amplitude terms, they also have a significant impacton the Doppler phase, that then should be compensated by modifying the source physicalparameters, λsrc.

At the end of the day, those five symmetry relations just can be combined in the followingtwo ways in order to leave the GW signal (2.38) invariant:

(a) : (2) + (4) −→ φ0 → π + φ0 ; ψ → π2 + ψ

(b) : (1) + (3) + (4) + (5) −→ ι→ π − ι ; ψ → π − ψ ; θn → π − θn(2.58)


III

IV I

II

ψ

φ

π

π

π

π

π

π

Figure 2.4: Diagram exemplifyinghow invariant transformation ‘(a)’,φ0 → π+φ0;ψ → π

2+ψ, is trans-

lated into equivalence between dif-ferent quadrants of the parameterspace, in particular I ↔ III andII↔ IV. [Plot: own production]

We recall that the initial phase, φ0, is defined in the [0, 2π)range, whereas the other four angles ι, ψ, θn are mod(π).Each of these two invariant transformations establish aone-to-one mapping between two halves of the parameterspace. For instance, in Fig. 2.4 we exemplify the invarianttransformation ‘(a)’, involving the angles φ0 and ψ. Anypoint in the quadrant I has its equivalent [i.e. that pro-duces the same h(t)] in the quadrant III and the samehappens between quadrants II and IV; thus we can saythat quadrants I - III are equivalent and also II - IV.In light of these results, it is obvious that carrying asearch over the entire parameter space, φ0 ∈ [0, 2π) andψ ∈ [0, π], would be inefficient, since we would be consid-ering all template waveforms twice.

Therefore, the invariant transformation ‘(a)’ allows us torestrict one of the angles involved to half its original range.This represents reducing the volume of the parameterspace [and also the computational time to explore it] by afactor 2. The same argument is applicable for ‘(b)’.

References 47

References

[1] B. Schutz, A First Course in General Relativity, Second Edition, (Cambridge University Press,Cambridge, England) (2009).

[2] T. A. Apostolatos et al., Phys. Rev. D 49 6274-6297 (1994).

[3] C. Cutler, Phys. Rev. D 57 7089-7102 (1998).

[4] L. E. Kidder, Phys. Rev. D 77 044016 (2008).

[5] A. Nagar, L. Rezzolla, Class. Quant. Grav. 22 R167 (2005).

[6] K. S. Thorne, Rev. Mod. Phys. 52 299-339 (1980).

[7] M. Trias, A. M. Sintes, Phys. Rev. D 77 024030 (2008).

[8] K. G. Arun et al., Phys. Rev. D 76 104016 (2007).


[10] E. K. Porter, N. J. Cornish, Phys. Rev. D 78 064005 (2008).

[11] M. Trias, A. M. Sintes, Class. Quant. Grav. 25 184032 (2008).

[12] P. Jaranowski, A. Krolak, B. F. Schutz, Phys. Rev. D 58 063001 (1998).

[13] M. Tinto, S. V. Dhurandhar, Living Rev. Rel. 8 4 (2005) .

[14] L. D. Landau, E. M. Lifshitz, Mechanics, Third Edition, (Butterworth-Heinemann, Oxford,England) (1976)

[15] T. Damour, N. Deruelle, Ann. Inst. Henri Poincare 44 263-292 (1986).

[16] J. H. Taylor, J. M. Weisberg, Astrophys. J. 345 434-450 (1989).

Chapter3Gravitational wave data analysisfor Compact Binary Coalescences

3.1 Gravitational wave signals buried in noise

The signal described in the previous chapter will be buried in the noise of a detector. Thus,we are faced with the problem of detecting the signal and estimating its parameters fromthe noisy observed strain. By assuming additive noise, the output strain of a detector,s(t), can be expressed as the sum of the GW signal characterized by the ‘true’ parametersλx, hx(t,λx) (here, and in the following, the subscript x stands for ‘exact’), and the noise,n(t);

s(t) = n(t) + hx(t;λx) . (3.1)

The noise is also assumed to be uncorrelated, stationary and with zero mean; and it ischaracterized by its (one-sided) power spectral density (PSD), Sn(f), which is defined as

n∗(f)n(f ′) =1

2Sn(f)δ(f − f ′) . (3.2)

Here, the overline denotes the ensemble average, the superscript star indicates complexconjugation, δ(f) represents the Dirac’s delta function and the tilde denotes the Fouriertransform,

n(f) =

∫ ∞−∞

n(t)e−2πiftdt . (3.3)

Note that we use here the LSC sign convention for the Fourier transform, with a factor∫dt e−2πift(...). This sign convention is what we shall use in Chapters 6 and 8; but opposite

to what we use in our parameter estimation studies including higher harmonics of theinspiral signal (Chapter 4). In the literature, there are also discrepancies between the LSCsign convention and the one used in many early GW papers (such as Refs. [1–3]) whichused the (theoretical-physics) convention

∫dt e+2πift(...).

According to this sign convention, the measured GW signal in the Fourier domain is usuallyrepresented in the following generic way,

h(f) ≡ A(f)e−iψ(f) , (3.4)

49

50 Chapter 3: GW data analysis for CBCs

where the amplitudeA(f) and the phase ψ(f) are real quantities. [Notice that the frequency-domain phase is denoted with the same symbol ‘ψ’ as the polarization angle; we shall alwayswrite the explicit frequency dependency of ψ(f) when referring to the frequency-domainphase.]

The assumed Gaussian and zero-mean properties of the noise, lead one to write that theprobability that an individual, ni, is a sampling of the random process n(t) is given by [4]

p(ni) = [2πCn(0)]−1/2 exp

[−1

2

(ni − 0)2

Cn(0)

],

where Cn(τ) is the correlation function of noise. Following the same arguments, the prob-ability that the ordered set n ≡ ni : i = 1, . . . , N is a sampling of n(t) will be

p(n) = [(2π)N det ||Cn,jk||]−1/2 exp

−1

2

N∑j,k=1

C−1n,jknjnk

where Cn,jk ≡ Cn[(j − k)∆t] and δjk ≡

∑iCn,jiC

−1n,ik; δjk being the Kronecker’s delta.

Taking into account that the previously defined (one-sided) noise PSD, Sn(f), is (twice)the Fourier transform of the correlation function, Cn(τ); Finn [4] shows that the previousexpression for the probability of having the realization of a particular noise series, n(t), inthe continuum limit of ∆t→ 0 and T →∞, can be written as

p(n) ∝ e−12 (n|n) , (3.5)

where the inner product (‘Wiener scalar product’) between two (real) time series, namelya(t) and b(t), has been defined as

(a|b) ≡∫ ∞−∞

dfa∗(f)b(f) + a(f)b∗(f)

Sn(f)= 2 Re

∫ ∞−∞

dfa∗(f)b(f)

Sn(f)= 4 Re

∫ ∞0

dfa∗(f)b(f)

Sn(f).

(3.6)In defining this inner product, we use the fact that the Fourier transform of the real time-domain gravitational wave strain h(t) satisfies h(f) = h∗(−f) [also, the one-sided noisePSD satisfies Sn(−f) = Sn(f)], thus allowing us to define the inner product as an integralover positive frequencies only. This Wiener scalar product is real and symmetric, and theassociated norm, say

|a|2 ≡ (a|a) = 4 Re

∫ ∞0

dfa∗(f)a(f)

Sn(f)= 4

∫ ∞0

df|a(f)|2Sn(f)

. (3.7)

is positive definite, and endows the space of (real) signals with an Euclidean structure. Alsonotice that the inner product is normalized such that, |n|2 ≡ 1; and, from the definitionof Sn(f) in Eq. (3.2), we get the useful property (a|n)(n|b) = (a|b).

In matched-filtering theory [5], the signal-to-noise ratio (SNR), ρ(λ), is defined as theratio between the filtered signal and the root-mean-squared (r.m.s.) of the filtered noise.Assuming ‘hm ≡ hm(λ)’ to be our best model template and using the definition of theWiener scalar product,

ρ(λ) =(hm|s)

r.m.s.[(hm|n)]=

(hm|s)√(hm|n)(n|hm)

=(hm|s)√(hm|hm)

≡ (hm|s) , (3.8)

3.2. Detection and parameter estimation 51

Here, we are using a hat to denote normalized templates h = h(h|h)1/2 , so that (h|h) ≡ 1.

Below, we shall see that, from a frequentist point of view, the optimal test statistics isgiven by the likelihood ratio (see Eq. (3.14) below); although Echeverria [6] showed thatmaximizing the likelihood ratio is equivalent to maximizing the SNR function definedabove, and therefore ρ(λ) = (hm|s) is an optimal search statistic.

We shall use the ensemble average of ρ(λ) to perform data analysis studies such as pa-rameter estimation, analyze template accuracies, or the computation of expected SNRs,

ρm ≡ ρ(λ) = (hm|s) = (hm|hx) +*

0

(hm|n) = (hm|hx) . (3.9)

The optimal SNR will be obtained when the template used hm is the exact signal hx thatwe want to detect, i.e.

ρopt = (hx|hx) =(hx|hx)

(hx|hx)1/2= (hx|hx)1/2 . (3.10)

3.2 Detection and parameter estimation

There are two different ways of proceeding from this point, depending on the paradigm ofstatistics used: frequentist or Bayesian. These approaches yield sometimes similar-lookinganswers for detection and parameter estimation (especially in Gaussian noise), but theyare based on fundamentally different interpretations and provide different tools in practice.The conceptual difference between the two frameworks lies in the meaning of “probability”,while the axioms for calculating with those probabilities are the same in both cases.

This section has been written following the guidance of several reference articles in thefield; in particular, the paragraphs discussing the parameter measurement accuracy arebased on Refs. [4, 7–9], whereas we use Refs. [10–13] for the discussion between frequentistand Bayesian frameworks.

3.2.1 Frequentist framework

The frequentist approach is based in viewing probabilities essentially as the relative fre-quencies of outcomes in repeated experiments: the probability p(A) of an event A is definedas the limiting fraction of events in a infinite number of “identical”1 trials.

Given the detector data s, consisting in the sum of the true signal hx = h(λx) [where the λxis the vector of the true system parameters] plus additive noise n; we select a point estimatorλ(s): that is, a vector function of detector data that (hopefully) approximates the truevalues of the source parameters, except for the statistical errors due to the presence of noise.We shall see below that an important point estimator is the maximum likelihood (ML)

estimator λml

, which maximizes the likelihood ratio p(s|hx)/p(s|0). The estimator λ isusually chosen according to one or more criteria of optimality: for instance, unbiasedness

requires that λ(s) (the ensemble average of the estimator) to be equal to λx.

1Obviously, the trials cannot truly be identical or they would always yield the same result.


Also, the statistical errors within the frequentist framework are characterized as the fluc-tuations of λ(s) computed over a very long series of independent experiments where thesource parameters are kept fixed, while the detector noise n is sampled from its assumedprobability distribution. Mathematically, this can be represented for a generic parameter,λi, as

varλi =(λi(s)− λi(s)

)2. (frequentist) (3.11)

The frequentist detection problem is formulated as one of hypothesis testing between H0

(no signal) or H1 (there is a signal). Then, a detection statistic Λ(s) is built and H0 isaccepted if Λ(s) < Λ∗; whereas if Λ(s) ≥ Λ∗ we accept H1. Of course, there may be caseswhere

• Λ(s) ≥ Λ∗, but there is no signal: false alarm probability, namely

α(Λ∗) ≡∫ ∞

Λ∗p(Λ(s)|H0) dΛ , (3.12)

which is the probability of a threshold crossing despite H0 being true; and

• Λ(s) < Λ∗, but there was actually a signal: false dismissal probability,

β(Λ∗|hx) ≡∫ Λ∗

−∞p(Λ(s)|H1) dΛ , (3.13)

which is the probability that the threshold is not crossed, even though H1 is true.The detection probability η is simply the complement, namely η(Λ∗|hx) = 1− β.

The test Λ(s) should maximize the detection probability, η, at a given false alarm rate,α. According to the Neyman-Pearson lemma, this optimal test is the so-called likelihoodratio, which is defined as

Λ(s;hx) =p(s|H1)

p(s|H0)=p(s|hx)

p(s|0)(3.14)

Since p(n) ∝ exp[−12(n|n)] and n = s− hx, we have that

p(s|hx) ∝ e−12 (s−hx|s−hx) and p(s|0) ∝ e−

12 (s|s)

and therefore, the optimal test within the frequentist framework can be written in termsof the inner product as

log Λ(s;hx) = (s|hx)− 12(hx|hx) , (3.15)

which is the well-known expression for the matched-filtering amplitude. Given a measureddata set s and the exact waveform model hx(λ) with unknown parameters, one has to findthe maximum of the log-likelihood function, log Λ, as function of the unknown parameters.With this, one has defined the maximum likelihood (ML) estimator, λ

ml.

The exploration of the parameter space searching for the maximum of the likelihood ratiois at the very heart of every GW search within the frequentist framework. When the


dimensionality of the parameter space is low and the computation of the likelihood ratio(3.15) is not very expensive in computational terms, one normally uses a uniform grid toplace the templates to be explored. When the problem is computationally more demanding(usually because of the high number of dimensions in the parameter space), differenttechniques have been explored in order to place the finite number of templates in the most‘interesting’2 parts of the parameter space, such as defining curved metric spaces [14–17],‘stochastic template banks’ [18–20], or using random template bank strategies [21].

As it has been previously mentioned, Echeverria [6] showed that maximizing the likelihoodratio (3.15) is equivalent to maximizing the SNR function (3.8), so both functionals canbe used as optimal search statistics.

It is needless to say that the threshold Λ∗ to determine whether a signal has been detectedor not, will be set, for each particular problem, by fixing the maximum allowed valuesgiven a false alarm and false dismissal probabilities, see Eqs. (3.12) and (3.13).

3.2.2 Bayesian framework

Bayesian statistics is built on a different concept of probability, quantifying the degree ofcertainty (‘degree of belief’) of a statement being true. So, one can assign probabilitiesp(A|M) ∈ [0, 1] to any statement A given a certain model M, quantifying one’s (possiblyincomplete) knowledge about the truth of A. In particular, for GW data analysis prob-lems in the Bayesian approach, we simply compute the posterior probability distributionfunction p(h|s,M) for the waveforms h of the model hypothesis M, and the associatedmodel evidence p(s|M). These are related by Bayes’ theorem to the likelihood p(s|h,M)of our observed strain s with our model waveform and the prior information p(h|M) as

p(h|s,M) =p(h|M) p(s|h,M)

p(s|M), (3.16)

the model evidence being

p(s|M) =

∫p(h|M) p(s|h,M)dh . (3.17)

Here, the likelihood function p(s|h,M) is the same as in the previous section, which [mak-ing use of the Gaussianity of the noise] is p(s|h,Mx) ≡ p(s|hx) ∝ exp[−1

2(s− hx|s− hx)]if one assumes that has access to the exact model. In reality, we only know an approxi-mation to the exact templates, and therefore the likelihood will be simply computed asp(s|h,M) ∝ exp[−1

2(s− hm|s− hm)].

Everything we want to know about the waveform model hm is contained in the posteriordistribution, while the evidence allows for comparisons to be made between alternativemodels (e.g. whether a GW signal is present or not; models with different number or kindof sources. . . ). It is important to emphasize that in the Bayesian approach, the output isonly as good as the inputs, in the sense that we are converting a prior knowledge aboutany statement into a posterior one throughout the likelihood of the measure s and ourtemplate waveforms h; but if the waveform model or the likelihood function is flawed,then the conclusion will also be flawed. The hope or expectation, both at the parameter

2With higher likelihood ratio values.


level and model level, is that data will be obtained of sufficient quality to overturn incorrectprior hypotheses.

The Bayesian evidence (3.17) sets up a tension between the ability of a model to fit the dataand the prior predictiveness3 of the model, in a quantitative implementation of Occam’srazor [22]. The models that do best are the ones that make specific predictions that laterturn out to fit the data well. Less predictive models, even if they can fit the data as well,score more poorly.

Any waveform model is characterized by a set of parameters (h|M) ≡ hm ≡ hm(λ), thusin Bayesian inference all the probability density functions (PDFs) shall be written as PDFsof the model parameters, which are taken as random variables: the priors p(h|M) ≡ p(λ);the likelihood function p(s|h,M) ≡ p(s|hm(λ)); and the posterior p(h|s,M) ≡ p(λ|s).Also, the integrals over h are, indeed, multidimensional integrals over all the parametersset, for instance the evidence (3.17) is computed as

p(s|M) =

∫p(λ) p(s|hm(λ))dλ . (3.18)

The output of the Bayes’ theorem is the joint PDF of all the parameters that characterizeour model λ = λ1, . . . , λN, but one can easily obtain the posterior PDF of a given sub-set of parameters, say λ1, . . . , λn with n < N , by marginalizing the original joint posteriorprobability

p(λ1, . . . , λn|s) =

∫dλn+1 . . .

∫dλN

p(λ) p(s|hm(λ))

p(s|M). (3.19)

With this, we can obtain the posterior PDF of any parameter or set of parameters and con-sequently, compute the posterior mean of a given parameter λi [we recall that in Bayesianinference, model parameters are taken as random variables],

〈λi〉p ≡∫λip(λi|s)dλi

/∫p(λ|s)dλ (3.20)

and the quadratic moment

〈λiλj〉p ≡∫λiλj p(λi, λj |s) dλidλj

/∫p(λ|s)dλ (3.21)

where “〈·〉p” denotes integration over the posterior PDF, p(λ|s). The variance varλi = σ2i

in the estimation of the parameter λi [σi being the root-mean-square of λi] can then becalculated as

varλi = σ2i = 〈(∆λi)2〉p = 〈(λi − 〈λi〉p)2〉p = 〈λ2

i 〉p − 〈λi〉2p , (Bayesian) (3.22)

and the correlation coefficients cij between two parameters λi and λj are given by

cij =〈∆λi∆λj〉p

σiσj=〈λiλj〉p − 〈λi〉p 〈λj〉p

σiσj. (Bayesian) (3.23)

3Which is not necessarily directly related to, for instance, the number of parameters.


Of course, equivalent expressions could be written for any higher moment. In case weare always considering the same model M, the associated model evidence will play thesimple role of a normalization constant in Eq. (3.16), and the same thing happens withthe integrals in the denominator of the previous equations: (3.20) and (3.21). Thus, wecan see how Bayesian parameter estimation can be naturally performed directly from theposterior PDFs, either by directly quoting them, or by computing parameter ranges withina certain confidence level, or just by computing variances.

The Bayesian detection problem turns out to be a simple model selection problem: wecompare the odds of the data being more consistent with our model of the gravitationalsignal and instrument noise, or our model of the instrument noise alone. This forces usto explicitly define an instrumental noise model and, if our particular model is poor – forinstance, not allowing for occasional glitches if they may happen – then the odds may favorthe detection hypothesis even when no signal is present in the data [12]. Again, Bayes’theorem can be applied to update a prior model probability by the evidence (3.18) andobtain the probability of the model M given the data s. In particular, since computingactual probability values for a certain model over all its competing models would requirea prior knowledge of all possible models testable by our experiment, which is impractical;what we compute is the odds ratio for one model over another:

O10 ≡p(M1|s)p(M0|s)

=p(M1)

p(M0)

p(s|M1)

p(s|M0)≡ P10 B10 . (3.24)

O10 is the odds ratio for models M1 and M0; an odds ratio & 3 favors M1 [23]. Theodds ratio has been written as the product of the prior odds P10 and the Bayes factorB10 (also known as the evidence ratio or marginal likelihood ratio). The former reflect anypreference between the models before the experiment is conducted, thus if we are to adoptone model over the other, the Bayes factor must overwhelm the prior odds, signaling thedata are sufficiently informative to distinguish between competing hypotheses.

Littenberg and Cornish [12] point out an important note of caution here, which is that aBayesian analysis is not answering the question “Is there a gravitational wave signal presentin the data, or is it just instrument noise?”, but rather, “Are the data most consistentwith our model of the gravitational wave signal and instrument noise, or our model of theinstrument noise alone?”.

Markov chain Monte Carlo

In typical GW data analysis applications the number of parameters needed to describe thesignal and noise models can be very large (ranging from tens to hundreds of thousands),and the regions of significant posterior weight generally occupy a minute fraction of thetotal prior volume. Resolving the peaks in the posterior distribution function requires avery fine sampling of the parameter space, but if this sampling is extended over the entireprior volume, the total number of samples can become astronomically large, discardingany possibility to sample posterior PDFs using grid-based methods.

The Markov chain Monte Carlo (MCMC) approach encompasses a powerful set of tech-niques for producing gridless samples from the posterior distribution, focussing their at-tention on regions of high posterior weight, and neatly avoiding the problem of computing


model evidence. The latter feature becomes less desirable when one is interested in modelselection and therefore, in computing Bayes factors. One solution is to generalize the tran-sitions between steps in the Markov chain (see below for a definition) to include “transdi-mensional” moves between different models, resulting in what are termed reversible jumpMarkov chain Monte Carlo (RJMCMC) algorithms. All the results presented in this thesis(see, in particular, Chapter 6) consider a single model, though the next step for futurework is to implement RJMCMC in the search for an unknown number of galactic binarysignals with LISA.

In general, MCMC methods are based on constructing a Markov chain that has the desireddistribution (target distribution, π) as its equilibrium distribution. The state of the chain(or value of the parameters λ describing the problem at hand) after a large number of stepsis then used as a sample from the desired distribution and, in particular, it can be used tocompute marginalization integrals such as the ones represented above [see e.g. Eqs. (3.17)-(3.21)] via Monte Carlo integration [24], when one fixes the target distribution to be thejoint posterior PDF of all parameters, π(λ) ≡ p(λ|s).

Metropolis-Hastings (MH) algorithms are employed in the overwhelming majority of thepractical implementations of MCMC methods in order to construct a Markov chain withthe desired properties. In MH updates, given a state of the Markov chain λ, one proposesa new state λ′ drawn from a proposal distribution (or transition kernel) q(λ,λ′). This newstate is accepted with probability

α(λ,λ′) = 1 ∧π(λ′) q(λ′,λ)

π(λ) q(λ,λ′)

, (3.25)

and the chain remains at λ with probability 1 − α(λ,λ′). In the previous equation, thenotation a ∧ b (for any real numbers a and b) stands for the minimum between a and b,and as a consequence the acceptance probability is limited to unity in the case where theratio is > 1. The iteration of this process constitutes the core for constructing a Markovchain whose equilibrium distribution, after an initial ‘burn-in’ period that is discarded inthe generation of the resulting PDFs, coincides with the target distribution, π; in our casebeing the joint posterior PDFs of all the parameters. A (renormalized) histogram of someof the parameter values characterizing the elements of the resulting chain represents theirposterior PDF, and an average along the sample path of the Markov chain of a function fof the parameters, will be an asymptotically unbiased estimator of

∫f(λ)π(λ)dλ. Let us

notice that MCMC algorithms prevent us to compute the model evidence since, for a fixedmodel, it only plays of a normalization constant. As we said, in case one were interestedin computing evidence ratios, then they should consider “transdimensional” moves andRJMCMC.

The convergence to the target distribution π is guaranteed through the acceptance prob-ability (3.25), but the convergence rate or efficiency of the chain is highly dependent onthe choice of the transition kernel q; indeed, its choice is one of the most critical stagesin actually implementing a MH-MCMC algorithm. If the proposed jumps are too short,the simulation moves very slowly through the target distribution; whereas if we jumpsare too big, most of the new proposed states lie into low-probability areas of the targetdistribution, causing the Markov chain to stand still most of the time. In a very interestingwork, Gelman et al. [25] study what are the most efficient [having defined ‘efficiency’ as therelative variance of an estimator from the MH-MCMC output compared to independentsampling] symmetric jumping kernels for simulating a normal target distribution using


the Metropolis algorithm. For a d-dimensional spherical multivariate normal problem, theoptimal symmetric jumping kernel has its scale ≈ 2.4/

√d times the scale of the target dis-

tribution and the acceptance rate of the associated Metropolis algorithm is approximately44% for d = 1 and declines to 23% as d→∞ [the efficiency of the Metropolis algorithm,compared to independent samples from the target distribution, is approximately 0.3/d].

In Chapter 6, we shall implement an algorithm based on MCMC in order to search for,first single and then multiple, galactic binary signals buried in LISA noise. On the way, weshall face the problem of a dramatic reduction in the convergence rate of the standard MH-MCMC implementation, due to the presence of multiple secondary maxima in the targetdistribution along several dimensions of the parameter space. This is an extended issuepresent in many MCMC applications. The solution proposed in Chapter 7 is a completelyMarkovian and fully general algorithm based on a technique called Delayed Rejection.

3.2.3 Fisher information matrix formalism

Over the last two decades, in the absence of confirmed GW detections, the prevailingattitude in the GW source-modeling community has been on exploring which astrophysicalsystems, and which of their properties, would become accessible to GW observations withthe sensitivities afforded by planned future experiments, with the purpose of committingtheoretical effort to the most promising sources, and of directing public advocacy to themost promising detectors. In this context, the expected accuracy of GW source parametersis often employed as a proxy for the amount of physical information that could be obtainedfrom GW observations. Currently, these kind of studies are being performed for the thirdgeneration ground-based detectors (e.g. the european Einstein Telescope), as well as thefuture interferometer space-based antenna, LISA.

Predicting the parameter-estimation performance of future observations is a complex mat-ter, mainly because there are a few analytical tools that can be applied generally to theproblem. In the source-modeling community, the analytical tool of choice has been theFisher information matrix (FIM), defined in terms of the Wiener scalar product as fol-lows,

Fij [h] ≡(∂h

∂λi

∣∣∣∣ ∂h∂λj). (3.26)

The FIM formalism has been extensively used in GW data analysis providing very usefulinformation about the expected physics that shall be obtained from GW observations;however, as Vallisneri points out in Ref. [10], one must be aware to not “abuse” of the FIMwhich could lead to misinterpretations. Vallisneri provides three different interpretations(all correct) of the meaning of the FIM and, in particular, of its inverse matrix F−1

ij [hx],that we quote here:

1. The inverse Fisher matrix F−1ij [hx] is a lower bound (generally known as the Cramer-

Rao bound) for the error covariance of any unbiased estimator of the true sourceparameters; thus it is a frequentist error.

2. The inverse Fisher matrix F−1ij [hx] is the frequentist error covariance for the max-

imum likelihood (ML) parameter estimator λml, assuming Gaussian noise, in thelimit of strong signals (i.e. high SNR) or, equivalently, in the limit in which the


waveforms can be considered as linear functions of source parameters [sometimescalled linearized-signal approximation (LSA)].

3. The inverse Fisher matrix F−1ij [hx] represents the covariance (i.e. the multidimen-

sional spread around the mode) of the posterior probability distribution p(λx|s) forthe true source parameters λx, as inferred (within the Bayesian framework) froma single experiment with true signal hx, assuming Gaussian noise, in the high SNRlimit (or in the LSA), and in the case where any prior probabilities for the parametersare constant over the parameter range of interest. Properly speaking, in this case theinverse Fisher matrix would be a measure of uncertainty rather than error, since inany experiment the mode will be displaced from the true parameters by an unknownamount due to noise.

As pointed out by Jaynes [26], while the numerical identity of these three different error-like quantities has given rise to much confusion, it arises almost trivially from the factthat in a neighborhood of its maximum, the signal likelihood p(s|λx) is approximated bya normal PDF with covariance Σij = F−1

ij . Let us recall that, given the covariance matrixΣij , the root-mean-square error σk in the estimation of the parameter λk can then becalculated by taking the square root of the diagonal elements of the covariance matrix,

σk = 〈(∆λk)2〉1/2 =√

Σkk , (3.27)

and the correlation coefficients cjk between two parameters λj and λk are given by

cjk =〈∆λj∆λk〉σjσk

=Σjk√ΣjjΣkk

. (3.28)

Here, the angle brackets denote an average either over noise realizations 〈·〉n if one considersthe frequentist interpretation, or over the posterior PDF 〈·〉p in the Bayesian framework.

In some cases, the Fisher matrix can be singular, so that the attempts to invert it nu-merically yield warnings that it is badly conditioned. In this case, the solution relies inremoving the eigenvectors with null eigenvalues from the Fisher information matrix, asthey represent combinations of parameters that leave the waveform unaltered. What ismore common in practice, is to have high correlations between parameters (without beingexact degeneracies) that give rise to ill-conditioned Fisher matrices, which inverse can becomputed, but may be difficult to. In particular, one can find an extreme sensitivity of theinverse Fisher matrix results to the small numerical errors in the input; which is why onemust be very cautious when computing the elements of the Fisher matrix: always trying todo as much of analytical calculations as possible, simplifying the expressions if possible andadopting higher-precision arithmetics. In particular, Fisher matrices computed from CBCswaveforms are frequently ill-conditioned4 and therefore one must be extremely careful inthe calculations (see Chapter 4). One way to check whether one is obtaining reliable resultsis to add small random perturbations, Monte Carlo-style, to the Fisher-matrix elementsand then verify the change in the covariance matrix.

In particular, in this thesis (Chapter 4) we have used the FIM formalism to study theimpact in the parameter estimation of adding the higher harmonics to the inspiral CBC

4Waveforms are characterized by a number of parameters which, in some places of the parameter space,can become degenerated.

3.3. Modeling gravitational waveforms from Compact Binary Coalescences 59

waveform observed with LISA. We also use these results to estimate the precision inmeasuring the dark energy equation of state with single LISA observations of SMBHscoalescence events (Chapter 5), i.e. we explore the repercussions of LISA observations incosmology. In all these studies, several actions have been carried out in order to preventthe possible effects from having ill-conditioned Fisher matrices and ensure the reliabilityof the FIM output results; in particular,

• the analytical expressions of the waveforms have been factorized in order to identifycommon factors and facilitate the derivation;

• all the waveform derivatives have been computed analytically; simplifying and fac-torizing the final expressions in order to eliminate possible numerical divergences;

• all the variables in the numerical code are “double precision” and the inversionprocess is done using the Cholesky decomposition implemented in gsl;

• the covariance matrix results have been generated over a grid in the parameter spaceand the output results present a smooth variation rather than strong oscillations;

• our results have been validated by the LISA Performance Evaluation (Taskforce)(LISA PE) [27].

Finally, let us recall that the FIM represents a fast and analytical tool to estimate thecovariance matrix under certain assumptions (high SNR or LSA) and it can be very usefulto obtain first approximations or to compare results when using different waveform models.However, results of posterior PDFs obtained within the Bayesian framework, despite beingmuch more expensive to obtain in terms of CPU time, will always be more faithful andrealistic than FIM. For extensive parameter estimation studies, normally what it is doneis to use the FIM formalism to explore all the parameter space and just compute Bayesianposterior PDFs for a very limited number of cases in order to check the validity of FIMresults.

Another important application of the FIM formalism is precisely to build the proposaljumps in MH-MCMC algorithms that sample posterior PDFs. As it is said in Sec. 3.2.2,for a d-dimensional spherical multivariate normal problem, the optimal symmetric jumpingkernel has its scale ≈ 2.4/

√d times the scale of the target distribution. The inverse Fisher

matrix can indeed provide an estimation of this scale beforehand.

3.3 Modeling gravitational waveforms from Compact Bi-nary Coalescences

During the last 5-10 years our knowledge of the dynamics of coalescing binary black hole(BBH) systems and their gravitational emission has experienced a huge progress, both withthe improvement of analytical methods, namely PN expansions and EOB approaches, andthe breakthroughs that occurred in Numerical Relativity (NR). The purpose of this sectionis to describe the main features and properties of such methods and to give references forfurther information. Since all the topics worked out in this thesis are focussed on dataanalysis (we have not contributed to the development of CBC waveforms), we shall present


here the different waveform models basically from an “end-user” perspective, withoutentering too much into details.

According to their validity range, work domain and method used to solve the dynamics;we can classify the different currently available model waveforms into several categories:

• time-domain PN models (either using closed-form expressions or integrating ordinarydifferential equations (ODEs)), which consist in expanding Einstein’s equations inTaylor series of some PN expansion parameter (usually a characteristic velocity, v)and solving them term by term. They provide a faithful description of the dynamicsin the inspiral part of the evolution, where the motion is still adiabatic. However,they fail describing the late-inspiral, merger and ring-down parts, where the velocitiesbecome highly relativistic;

• closed-form, frequency-domain models which are only accurate in a limited rangeof the evolution. Into this category would fit either the analytically Fourier trans-formed PN models (using the Stationary Phase Approximation, see Sec. 3.3.1) andthe (NR-fitted) phenomenological models, which try to extend their validity rangeby combining information coming from closed-form PN (for the inspiral part) andringdown models, and some NR simulations (for the merger part and in order to linklate-inspiral and ring-down);

• EOB models, which incorporate information from both PN theory and NR simu-lations into an accurate description of the full dynamics of a CBC, from the earlyinspiral up to the end of the merger. They are defined in the time-domain by inte-grating ODEs;

• full NR simulations, which are necessary for describing the nonperturbative physicsaround the merger, but can only cover a limited number of orbits (. 15) becausethey are very time consuming in terms of computational costs;

• various types of hybrid models, joining together (either in the time-domain, or in thefrequency-domain) early PN-type waveforms to later NR waveforms, so as to coverthe whole evolution.

The time needed to generate a waveform in one of these categories varies by many orders ofmagnitude between, say, a closed-form frequency-domain waveform (a fraction of a secondon a single CPU); an EOB waveform (several seconds, also, on a single CPU); or, a fullNR simulation (∼ a month on a computer cluster with several hundreds of CPUs). Also, ofcourse, the accuracy of the waveforms produced by each of these models varies inversely;having the NR waveforms to be the most accurate (one could indeed think that the exactwaveform would be contained within the numerical errors), then the EOB ones, and finallythe less accurate are the closed-form waveforms (either PN or phenomenological). Fromthe data-analysis point of view, it would be quite useful to be able to generate waveformsas fast as possible, though the real issue is to be able to access a dense bank of themvery fast. Thus, though some trade off between fastness and accuracy can be allowed for,there are minimal accuracy requirements that waveforms have to satisfy. In Chapter 8 weperform a detailed analysis of the accuracy of the fastest existing waveforms, namely theclosed-form, frequency-domain ones.


Let us now summarize the recent progress made on the different approaches and point outsome references for further information. Finally, in Sec. 3.3.1, we shall describe an ana-lytical (and approximated) procedure to obtain Fourier transformed expressions of time-domain chirping signals [such as the inspiral part of CBC waveforms], called StationaryPhase Approximation (SPA).

Post-Newtonian models

The post-Newtonian approximation computes the evolution of the GW phase φgw(t) (twicethe orbital phase φorb, see Chapter 2) of a compact binary as a perturbative expansion in asmall parameter, typically taken as the characteristic velocity in the binary, v = (πMF )1/3,or x ≡ v2, although other variants exist. We recall that M = m1 + m2 is the total massof the binary and F = φgw/(2π) is the GW frequency. PN evolution is based on the so-called adiabatic approximation, according to which the fractional change in the orbitalfrequency over each orbital period is negligibly small, i.e. Forb/F

2orb = Fgw/(2F

2gw) 1.

This assumption is valid during most of the evolution, but starts to fail as the systemapproaches the last stable orbit (LSO), where vlso = 1/

√6.

Given PN expansions of the motion of, and gravitational radiation from, a binary system,one needs to compute the “phasing formula” for φgw(t;λsrc). In the adiabatic approxima-tion, the phasing formula is easily derived from the energy and flux functions by a pair ofdifferential equations,

φgw(t) = 2v3/M and v = −F(v)/E′(v) ,

where F(v) is the GW luminosity and E′(v) is the derivative of the binding energy with re-spect to v. Different PN families arise because one can choose to treat the ratio F(v)/E′(v)differently starting formally from the same PN order inputs [3]. For instance, one can re-tain the PN expansions of the luminosity F(v) and E′(v) as they appear (the so-calledTaylorT1 model), or expand the rational polynomial F(v)/E′(v) in v to consistent PNorder (the TaylorT4 model [28]), recast as a pair of parametric equations φgw(t) and t(v)(the TaylorT2 model), or the phasing could be written as an explicit function of time φ(t)(the TaylorT3 model). These different representations are made possible because one isdealing with a perturbative series. Therefore, one is at liberty to “resum” or “re-expand”the series in any way one wishes (as long as one keeps terms to the correct order in theperturbation expansions), or even retain the expression as the quotient of two polynomialsand treat them numerically. There is also the freedom of writing the series in a differentvariable, say (suitably dimensional) E (the so-called TaylorEt model [29–31]).

The description of CBCs through PN expansions theory has been developed for manyyears and currently, the evolution of the orbital phase is known up to 3.5PN order (i.e. upto O(v7) in the PN expansion) for non-spinning systems in circular orbits [3, 32–39];see, in particular, Ref. [3] for a complete summary of useful expressions. Moreover, theamplitude of the dominant (2, 2) mode can be computed up to 3PN order [40], whereasthe remaining components (up to ` = 8) are also known up to lower PN orders (see alsoRef. [40] for explicit expressions). Also, a lot of work has been done describing the spineffects within the PN approximation [32–34, 41–48]; in particular, we know the leadingand next-to-leading order spin-orbit effects [42–44] and the spin-spin effects that appearat relative 2PN order [44–46], these latter only valid when the compact objects are BHs.The amplitude corrections for spinning objects are presented in Ref. [47, 48].


By combining PN expansions with an approximated way to compute the Fourier transform(FT) of a chirping signal, namely the SPA (see below, Sec. 3.3.1), one can obtain closed-form expressions directly in the Fourier domain. Here too, there are many inequivalent waysin which the phasing ψ(f) [see Eq. (3.4)] can be worked out; the most popular ones consisteither in substituting for the energy and flux functions their PN expansions without doingany re-expansion or resummation (the TaylorF1 model), or in using the PN expansionsof energy and flux but re-expand the ratio F(v)/E′(v) (the so-called TaylorF2 model) inwhich case the integral can be solved explicitly, leading to an explicit, Taylor-like, Fourierdomain phasing formula: ψ(f) = 2πftref − φref + 3

128ηv5

∑Nk=0 αkv

k [see Ref. [49] for the

explicit expression of the αk coefficients up to 3.5PN order]. The terms tref and φref aresimple reference time and phase values.

Numerical Relativity simulations

Since the initial breakthroughs in 2005 [50–52], there has been dramatic progress in NRsimulations for GW astronomy, including many more orbits before merger, greater accu-racy and a growing sampling of the BH binary parameter space. There are now severalgroups around the world with codes capable of performing BBH simulations [51–62]. Thereare two analytic forms of the Einstein equations (BSSN [63–65] and the Generalized Har-monic [51, 66, 67]), and various numerical methods (high order finite differencing, pseudo-spectral, adaptive mesh refinement, multi-block) in use in the community. A summaryof the published “long” waveforms is given in the review [68], and a complete catalog ofwaveforms is being compiled in [69]; more recent work is summarized in [70]. NR resultsare now accurate enough for GW astronomy applications over the next few years [71], andhave started playing a role in GW searches [72, 73].

Accurate NR simulations will produce the most accurate and faithful results of any of themethods we summarize here. However, the computational cost to perform a single NRsimulation is about a month on a computer cluster with several hundred of CPUs (thisis a factor ∼ 108 longer [in actual CPU time] than EOB; and much more compared toclosed-form frequency domain waveforms) and the longest NR BBH waveform producedso far lasts for 15 orbits, includes the merger and ringdown phases and corresponds to anequal-mass case [74, 75]. Moreover, for a 3D NR code, one of the most challenging problemsis the simulation of BHs of very different masses, and the reason is that, for a fixed totalmass M = m1 + m2, the gravitational wavelength remains approximately constant withvarying the mass ratio q, but the length and time scales required to resolve the smallerhole scales approximately with q. Indeed, the highest mass ratio that has been simulatedis q = 10 [70].

Thus, there are several practical limitations that avoid NR waveforms to be extensivelyused for GW astronomy, namely (i) the extremely high computational cost, (ii) number oforbits . 15 and (iii) mass ratios q ≤ 10. For this reason, different groups are working onseveral approaches to make use of the information from NR simulations in order to buildapproximated, but much faster, models (see below).


Effective-One-Body approach

The EOB formalism [76–78] has the unique feature of being able to incorporate informationcoming both from PN theory and NR simulations into an accurate description of the fulldynamics and GW radiation of a CBC, from the early inspiral up to the end of themerger. In particular, besides incorporating the full PN information available, the EOBformalism goes beyond PN theory, even during inspiral, in several ways: (i) it replaces thePN-expanded results by resummed expressions that have shown to be more accurate invarious cases; (ii) it includes more physical effects, notably those associated with the nonadiabatic aspects of the inspiral. We recall that all the PN models presently considered inthe literature make the approximation that the GW phase can be computed in an adiabaticmanner, by simply using the balance between the GW energy flux, F(v), and the adiabaticloss of binding energy of the binary system, E(v), (see above). In addition to improvingany PN model (even in the inspiral part), the EOB formalism naturally introduces theleading order of the unknown parameters to be calibrated using a very reduced number ofNR simulations. All these improvements have made the currently most accurate versionof the EOB waveforms [79] to agree, within the numerical error bars, with the currentlymost accurate NR waveform [74, 75, 80].

Originally, the EOB formalism was introduced by Buonanno and Damour in Ref. [76] as anextension to the non-adiabatic regime of the inspiral and right after, they were able to addthe ring-down phase [77] and therefore to have a representation of the full dynamics, fromthe early inspiral up to the end of the merger. The extension of the EOB Hamiltonianto spinning systems was first done in Ref. [46], which then was enhanced by includingnext-to-leading order spin-orbit coupling effects [81].

Over the last years, it has become possible to improve the EOB waveform by comparing itto the results of accurate NR simulations and by using the natural ‘flexibility’ of the EOBformalism to ‘calibrate’ some EOB parameters representing either higher-order perturba-tive effects that have not yet been analytically calculated, or non-perturbative effects thatcan only be accessed by NR simulations [79, 80, 82–85]. A similar work has recently beendone using some spinning, non-precessing and equal-mass NR simulations [86]. In addi-tion, several theoretical improvements have been brought in the EOB formalism, notablyconcerning new ways of resumming the GW waveform [83, 87], and the GW radiationreaction [79].

Phenomenological models

The coalescing process of a compact binary system is divided into three consecutive phases:inspiral, merger and ringdown. We have seen that PN models are accurate describing theadiabatic inspiral phase of the evolution which covers the major part of the coalescence:from the beginning of the inspiraling process up to several orbits before the merger. How-ever, it is during these very last orbits (late inspiral – merger – ringdown phases) wheremost of the (GW) luminosity is emitted and therefore, they play a crucial role for GWdetection. An accurate description of these very last, highly relativistic orbits in the coales-cence process is obtained from NR simulations, which provide the most faithful descriptionof the actual dynamics but they just can cover a limited number of simulated orbits.


When there exists a frequency range in the late inspiral phase, where (i) PN models are stillaccurate enough and (ii) it is close enough to the merger so that NR simulations alreadycover it; then one can build hybrid waveforms and obtain a waveform valid in all the stagesof the evolution. [These hybrid waveforms are also build even when one is working withEOB and wants to have access to the very low frequency part of the waveform, in the deepinspiral, where the evolution is so slow and adiabatic that there is absolutely no need touse anything else than PN (see Chapter 8).]

Given the computational cost of producing NR simulations, the number of hybrid wave-forms that one has access to is finite. What Ajith et al. [88, 89] and Santamaria et al. [90]propose is to take a discrete sample of these hybrid waveforms in the frequency domainand build closed-form phenomenological waveform models, i.e. using the known functionalforms that better represent the different stages (e.g. a PN expansion for the GW phaseand inspiral part of the GW amplitude, or a Lorentzian function for the amplitude duringthe ring-down), but considering the phenomenological coefficients that best matched eachof the original hybrid waveforms. A simple polynomial fitting of each coefficient along thedifferent dimensions of the parameter space completes the construction of the closed-formphenomenological model. At the end, A(f) and ψ(f) are functions of the phenomenologicalparameters, which at the same time are given in terms of the physical parameters (massratio, spins, . . . ).

3.3.1 Stationary Phase Approximation

Einstein’s equations are usually solved in the time domain and so are expressed the re-sulting waveforms. However, data analysts normally work in the frequency domain as thereader must have noticed from all the derivations made above in Secs. 3.1 and 3.2 [thereason relies in that most of the GW signals are quasi-monochromatic and so they canbe expressed in a more compact way in the frequency domain]. Thus, the first step inany data analysis study will consist in Fourier transforming the time-domain waveforms.Normally, this can be done very fast using the Fast Fourier Transform (FFT) [91], whichis an efficient numerical algorithm to compute the discrete version of the FT expressedin Eq. (3.3) in O(N logN) instead of the O(N2) arithmetical operations that would berequired by making use of its raw definition,

xk =N−1∑j=0

xj e−i2π jkN where

xi ≡ x(ti)

ti = i∆tand

xk ≡ 1

∆t x(fk)

fk = kTobs≡ k

N∆t ≡ k∆f.

(3.29)

Indeed, the FFT is what shall be used to obtain EOB or NR waveforms in the frequencydomain, because the integration process to obtain the time-domain waveforms is muchlonger than the FFT, and also because these models are normally used to simulate thelate-inspiral, merger and ringdown parts of the evolution, where the dynamics are highlyrelativistic. However, PN waveforms are a whole different story: (a) they can be writtenas closed-form Taylor series, making their evaluation process extremely fast; (b) they arevalid from t → −∞ up to a certain point before the last stable orbit, which means that(i) the dynamics are in an adiabatic regime and (ii) one can generate extremely longPN waveforms starting from the deep-inspiral that will require a huge number of datapoints to proper sample them in the time-domain. These two reasons make the FFT (for


PN waveforms) a slow process in comparison to the waveform’s evaluation time and it ishighly recommended to look for an analytical alternative.

The Stationary Phase Approximation (SPA) is a simple, explicit, analytic approximationto the FT of a highly oscillating time-domain signal, such as a chirp. More precisely theSPA is valid when the amplitude and frequency of the signal evolve on time scales muchlonger than the orbital period: εa = a(t)/(a(t)φ(t)) 1 and εf = φ(t)/φ2(t) 1. For acoalescing binary system, the time scale of evolution of the GW amplitude and frequencyis the radiation-reaction time scale so that εa ∼ εf ∼ νv5 stays much smaller than 1essentially up to the end of the inspiral.

The SPA is as follows. Let us consider a generic time-domain oscillating signal,

h(t) = a(t) cosφ(t) =1

2a(t)(eiφ(t) + e−iφ(t)) , (3.30)

where φ(t) changes in a time scale much shorter than a(t) and φ(t) ≡ 2πF (t). Then, itsFT can be computed as [see Eq. (3.3)]

h(f) ≡∫ ∞−∞

dt h(t)e−i2πft =

∫ ∞−∞

dta(t)

2

(ei(φ(t)−2πft) + e−i(φ(t)+2πft)

). (3.31)

The integrands of h(f) are violently oscillating and therefore, their dominant contributionscome from the vicinity of the stationary points of their phase (when such points exist).

When F (t) = φ(t)2π > 0 (which we shall henceforth assume), only the ei(φ(t)−2πft) ≡ eiξ(t)

term has such saddle point,

ddt(φ(t) + 2πft) = 0 ⇒ φ = −2πf ⇒ 2πF (t) = −2πf ⇒ F (t) = −f

[ not possible

F (t)>0

],

ddt(φ(t)− 2πft) = 0 ⇒ φ = 2πf ⇒ 2πF (t) = 2πf ⇒ F (t) = f [→ tf ] .

By expanding a(t) and ξ(t) ≡ φ(t)− 2πft around the saddle point, tf [with ξ(tf ) ≡ 0],

a(t) = a(tf ) + a(tf )(t− tf ) + · · · , (3.32)

ξ(t) = ξ(tf ) +

ξ(tf )(t− tf ) + 1

2 ξ(tf )(t− tf )2 + · · · , (3.33)

we can finally write the well-known expression for the usual SPA:

h(f) ≈∫ ∞−∞

dt 12 [a(tf ) + a(tf )(t− tf )] ei[ξ(tf )+

12 ξ(tf )(t−tf )2] [t−tf≡t]

=

=1

2eiξ(tf )a(tf )

∫ ∞−∞

dt ei12 ξ(tf )t2 +

1

2eiξ(tf )a(tf )

:0∫ ∞

−∞dt t ei

12 ξ(tf )t2

=1

2eiξ(tf )a(tf )

√π

|2πF |(1 + i)

[ξ=φ−2πft]=

=1

2√F (tf )

a(tf )e−i(2πftf−π4−φ(tf )) , (SPA) (3.34)

where we recall that tf is defined as the saddle point where F (tf ) = f .

For GW signals emitted by CBCs, the signal’s phase consists in the sum of the GW phaseφgw, plus the Doppler ϕd and polarization ϕp phases [see Eq. (2.52)]. This also means,that F (t) shall have contributions from the time derivatives of the three components.


The generalization of Eq. (3.34) to a time-domain signal consisting in the sum of severalharmonics is straightforward. Let us write the generic time-domain expression as in (2.55),i.e. in terms of a signal’s phase φ(t) twice the orbital phase [also the signal’s frequency is

F (t) = φ2π = 2forb]; hence the signal h(t) shall be written as

h(t) =∑j

aj(t) cos j2φ(t) . (3.35)

By applying the same arguments as before, the generic SPA expression for a time-domainsignal with multiple harmonics can be written as

h(f) ≈∑j

1

2√

j2 F (t

(j)f )

aj(t(j)f ) e

−i(

2πft(j)f −

π4−

j2φ(t

(j)f )

) , (SPA) (3.36)

where it is important to note that t(j)f is different for each harmonic, j, and can be com-

puted as F (t(j)f ) = 2f/j. Also notice that in Eq. (3.35) we have included all the phase

contributions into j2φ(t); by comparison to Eq. (2.55) of a CBC gravitational waveform,

we have that

φ(t) ≡ φgw + 2jϕp,j + 2

jϕd . (3.37)

From the definition of F (t) = φ2π we could also obtain its dependency in terms of the

physical quantities defined in Chapter 2.

Finally, let us emphasize that this derivations correspond to the SPA just up to the leadingorder, which is more than enough for all the practical purposes that we shall consider inthis thesis. However, Damour et al. in Ref. [2] [also Droz et al. [92]] go beyond the leadingorder and show that by keeping two more terms in both Taylor expansion of Eqs. (3.32)and (3.33), the new SPA expression is simply the same as in (3.34) with a phase correctingfactor eiδ, where

δ ≡ 1

2πF (tf )

−1

2

a

a+

1

2

a

a

F

F+

1

8

...F

F− 5

24

(F

F

)2t=tf

. (3.38)

3.4 Visualizing gravitational waveforms from CBCs

The coalescing process of a compact binary system, when it is still far away from themerger consists in a slow, adiabatic rotation of each of the compact objects around thecenter of mass of the system. As they approach each other due to the energy emission[mainly in form of GWs], the orbital frequency increases as well as the amplitude of theemitted gravitational signal. Finally, the two compact objects merge to form a singleexcited Kerr black hole (if their total mass is high enough), which shall still be emittingGWs in a form of quasi-normal modes until it reaches the fundamental state of a KerrBH. The resulting gravitational waveform emitted in the overall process can be describedby a highly oscillating ‘chirping’ signal [the signal of the dominant mode oscillates with afrequency twice the orbital frequency], i.e. whose amplitude and frequency increase withtime.

3.4. Visualizing gravitational waveforms from CBCs 67

5

10

15

30

210

60

240

90

270

120

300

150

330

180 0

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

2

4

6

8

10

12

14

t × 100MS

M

r/M

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00

10

20

30

40

50

60

70

t × 100MS

M

ϕorb

Figure 3.1: Trajectory of the last 10 orbits of an equal-mass, non-spinning BBH system in polar co-ordinates obtained from an EOB simulation. The right panels represent the time evolution of the twopolar coordinates (r,ϕorb). [Plots: own production, using an EOB simulation courtesy of A. Nagar andT. Damour]

−250 −200 −150 −100 −50 0

−0.4

−0.2

0

0.2

0.4

Re[h

22(t)]

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0

−0.4

−0.2

0

0.2

0.4

Re[h

22(t)]

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

−0.4

−0.2

0

0.2

0.4

Re[h

22(t)]

1

t × 100MS

M

0

0.1

0.2

0.3

0.4

|h22(t)|

4100

4300

4500

4700

phase,ε22(t)

−5 −4 −3 −2 −1 00

0.02

0.04

0.06

0.08

0.1

ε22(t)

2π

×M

t × 100MS

M

Figure 3.2: Time-domain gravitational waveform emitted by an equal-mass, non-spinning BBH system(see also Fig. 3.1). Left panels represent snapshots of the real part of h22(t) [see Eqs. (2.2) and (2.10)] atthree different time range levels; whereas the right panels show the amplitude, phase [see Eq. (2.12) fordefinition and discussion of ε`m(t)] and phase derivative of its (2, 2) mode as a function of time. [Plots:own production, using an EOB simulation courtesy of A. Nagar and T. Damour]


10−1

100

101

102

10−1

100

101

102

103

104

A22(f

)

f × M

100MS

1500 orbits15 orbits

Merg

er

1 sec to merger

(@100 Ms)

5 sec1 min1 hour1 day1 week

10−1

100

101

102

−7

−6

−5

−4

−3

−2

−1

0

1x 10

4

ψ22(f

)

f × M

100MS

1500 orbits15 orbits

Merg

er

1 sec to merger

(@100 Ms)

5 sec1 min1 hour1 day1 week

Figure 3.3: Frequency-domain gravitational waveform emitted by an equal-mass, non-spinning BBH sys-tem (see also Figs. 3.1 and 3.2); in particular, we write h22(f) ≡ A22(f)e−iψ22(f) [see Eq. (3.4)]. Theoverprinted segments should help visualizing how ‘time’ (for M = 100M) and ‘number of orbits’ rangestranslate into the frequency-domain. [Plots: own production, using an EOB simulation courtesy of A. Nagarand T. Damour]

100M 1s 1min 1h 1d 7d 10−1 Hz 102 Hz

10M 0.1s 6s 6min 2h 24min 16h 48min 1 Hz 1000 Hz103M 10s 10min 10h 10d 2m 9d 0.01 Hz 10 Hz105M 16min 40s 16h 40min 1m 11d 2yr 9m 19yr 2m 10−4 Hz 0.1 Hz107M 1d 4h 2m 8d 11yr 5m 275yr 1.9kyr 10−6 Hz 10−3 Hz109M 3m 24d 19yr 1.15kyr 27kyr 192kyr 10−8 Hz 10−5 Hz

Table 3.1: All the plots included in Figs. 3.1, 3.2 and 3.3 use the M = 100M system as reference,i.e. we plot t× 100M

Mand f × M

100M. This table “translates” some of those ‘time’ and ‘frequency’ values

for different M , keeping t/M and fM fixed. Note that heavier systems have longer time scales and loweremission frequency.

3.4. Visualizing gravitational waveforms from CBCs 69

In general, the intrinsic parameters that characterize a BBH coalescence are the total massof the system M , the mass ratio q and the two spins of the BHs S1 and S2. In case ofconsidering NSs or other compact objects with presence of matter, then one has to alsoconsider some extra parameters characterizing its equation of state. In this thesis, we onlyconsider non-spinning BHs (or NSs far away from the merger, where the equation of stateparameters are irrelevant) and therefore, the two only intrinsic parameters will be M, q.Moreover, it turns out that after writing Einstein’s equations in their adimensional form,the total mass M only plays the role of a scale factor, and therefore one considers5 anadimensional time, t/M , an adimensional frequency, fM , etc. This means that a certainprocess that happens in a given amount of (adimensional) time t/M , will be slower inactual ‘seconds’ as the total mass of the system increases. The opposite happens for the(adimensional) frequency; this is, heavier systems emit at lower frequencies (see Tab. 3.1and the explanation below for further details).

Figures 3.1, 3.2 and 3.3 graphically represent all the relevant quantities of a CBC process,both, in the time and frequency domains. The results have been generated for a non-spinning, equal-mass system; using the EOB approach [87] to generate the last 1500 orbitsprior the merger, plus the merger and ring-down phases [the simulation took ≈ 500 secrunning on a single core of a 2.40 GHz Intel Core 2 Duo desktop machine with 2 Gb ofRAM]; and joining6 in front of it the PN-SPA result, which is perfectly valid ∼ 1000 orbitsbefore the merger. Assuming a fiducial M = 100M, the last 10 orbits plotted in Fig. 3.1would last 1 sec and, as it can be seen in Fig. 3.3, the merger would be emitted at around100 Hz. Taking into account the fact that the total mass M simply plays the role of a scalefactor in Einstein’s equations, this time and frequency values can be easily converted toother M values, and this is indeed what has been done in Tab. 3.1. Notice that observablemergers (say, from 15 orbits before merger up to the end) in the LIGO frequency band(10 − 104 Hz) will correspond to systems with a total mass M ∈ [2, 100]M and theywill last less than 1 second; considering the LISA band (10−5 − 10−1 Hz), on the otherhand, the systems will have a mass M ∈ [2× 105, 108]M and the events will last at least30 minutes and typically more than a day. These numbers are important to understandsome of the approximations made in the previous sections, such as assuming the ground-based detectors antenna patterns (whose variation is due to the Earth’s motion) to beconstant during the observation time of a merger (see Sec. 2.2.2).

The dependency of the emitted waveforms with the mass ratio q is not straightforward, andone has to run a new simulation every time that changes q. Qualitatively, given a certainseparation distance between the compact objects, the evolution is slower [and so is largerthe number of orbits before merger] as the mass ratio increases. Figure 3 of Chapter 8shows an example of amplitudes and phases in the frequency domain of EOB waveformsfor q = 1, 2, 4, 10 cases.

Also notice, in the right panels of Fig. 3.2, that the most important changes both infrequency and amplitude are produced in the last instants before merger, however theevolution of these two quantities during the whole inspiral stage up to several orbits beforemerger remains very slow. This slow chirping of the signal during most of the time of theevolution makes the frequency domain representation very useful, especially because of itscompactness: note, for instance, the simple behavior of A22(f) and ψ22(f) in Fig. 3.3 at

5We recall that we are always working with units such that G = c = 1.6See Section IV of Chapter 8 for further details.


low frequencies and the high number of orbits (and time) that this part represents. Indeed,for data analysis purposes one usually works in the frequency domain.

Although the time-domain amplitude of a chirping signal increases with time, this is notnecessarily the case in the frequency domain (see left panel of Fig. 3.3). The reason isbecause the frequency-domain amplitude represents the amount of energy accumulated ina certain frequency bin; thus, even if one has lower amplitude values in the deep inspiralstage, since the frequency evolution is also much slower, the system remains in that fre-quency bin for many more orbits, accumulating more energy and therefore having higherfrequency-domain amplitudes.

3.5 Effective/characteristic amplitudes and observable sources

In the previous section we have represented and analyzed all the relevant quantities char-acterizing the gravitational waveform emitted by a CBC (see Figs. 3.1-3.3). We have ob-served how the frequency-domain representation of the waveform is a much more compactand simpler way to visualize the emission, although one always have to have in mind thehighly non-linear relation that exists between the time and frequency. Thus, for instance,a M = 100M system needs just 1 second (15 orbits) to increase their emission frequencyby an order of magnitude (from 10 Hz to 100 Hz) when it is close to the merger, but itneeded ≈ 1 hour (1500 orbits) to go from 1 Hz to 10 Hz. These facts become importantwhen one considers finite observation times. The aim of this section is to discuss aboutthe best way to represent the signals’ amplitudes together with the detector’s sensitivitycurves, and quickly get an idea of the expected SNR that a certain signal observed witha given interferometer would have; this is, we shall talk about effective and characteristicamplitudes.

This topic was addressed many years ago [2, 93] and several authors proposed differentdefinitions for the effective amplitude depending on the nature of the signal; or, in partic-ular, depending whether the signal is observed as a continuous wave (i.e. Tobs signal’sduration) or a burst-like signal (i.e. Tobs signal’s duration). Since signals from CBCscan be seen in either ways depending on the total mass, luminosity distance and detector,we shall consider here a combination of the various “effective” amplitudes defined in theliterature.

The main purpose in defining effective quantities is that they have to provide intuitiveinformation about the observed SNR. Let us start recalling which is the definition of theoptimal SNR (squared) given a certain waveform, h, in terms of the frequency domainsignal and the noise PSD [see Eq. (3.10)],

ρ2opt = (h|h) = 4

∫ F (t0+Tobs)

F (t0)df|h(f)|2Sn(f)

=

∫ F (t0+Tobs)

F (t0)

df

f

f2|h(f)|2f4Sn(f)

. (3.39)

Here, we have explicitly written the integration intervals for a given finite observationaltime, Tobs, starting the observation at t0, and F (t) being the GW frequency [here we onlyconsider the (2, 2) mode, though the extension to higher harmonics would be straightfor-ward]. Moreover, on the r.h.s. of Eq. (3.39) we have rewritten the integral expression ofthe optimal SNR so that each of the terms are adimensional:

3.5. Effective/characteristic amplitudes and observable sources 71

• First, we have written the integral over the (adimensional) logarithmic frequency,dff = d log f , so that in the usually represented log-scale plots, integrals shall be

intuitively computed as the ‘area below the integrand’.

• Despite the time-domain GW amplitude, h(t), being adimensional [as it representsthe relative length changes of the two interferometer arms, see Eq. (2.32)]; the defini-tion of the Fourier transform, Eq. (3.3), adds units of time to the frequency-domainsignal, h(f). Hence, f2|h(f)|2 is adimensional and we shall define it as the (squared)effective gravitational signal [2],

h2s(f) ≡ f2|h(f)|2 . (effective signal) (3.40)

• Finally, the detector noise PSD, Sn(f), has also units of time, which will cancel outwhen it is multiplied by f . The factor 1

4 comes from the fact that we are consideringone-sided noise PSDs [the two-sided noise PSD would have just a factor 1

2 ]. Thus,we define the effective GW noise [2] (or, also called characteristic noise amplitude[93]) as

hn(f) ≡√

f4Sn(f) . (effective noise or characteristic noise) (3.41)

With these definitions, the optimal SNR (squared), given in Eq. (3.39), can be writtensimply as the logarithmic frequency integral of the squared ratio of the (adimensional)effective signal and noise,

ρ2opt =

∫ F (t0+Tobs)

F (t0)

df

f

h2s(f)

h2n(f)

. (3.42)

Now, the usefulness of the ratio hshn

as indicator of the SNR will depend of the frequencyvariation during the observational time. If one considers a CBC signal in the last stagesof the evolution (with a significant frequency variation) and observed for a sufficient longtime so that the frequency changes by, at least, one of order of magnitude, then the ratioof effective signal and noise will provide an intuitive idea of the optimal SNR. However,if one considers a quasi-monochromatic signal (e.g. a CBC in the deep inspiral phase),then the frequency variation during Tobs will be negligible, so the integrand will be almostconstant and the result of the integral in Eq. (3.42) will basically be ∆(logF )× ( hshn )2 with

∆(logF ) ≡ log FendF0 1 and therefore ( hshn )2 ρ2

opt.

For cases like the latter, namely ∆(logF ) 1; it is far more useful to work with char-acteristic quantities [93], rather than the effective ones defined in (3.40). In particular,the characteristic noise is equal to the effective noise [see Eq. (3.41) above] and since weare integrating it in a narrow frequency band, it can be taken as constant and factorizedout of the integral. The remaining part is defined as the (squared) characteristic signalamplitude,

h2c(f) ≡

∫ F (t0+Tobs)

F (t0)

df

fh2s(f) =

∫ F (t0+Tobs)

F (t0)

df

ff2|h(f)|2 . (characteristic signal)

(3.43)Then, the characteristic frequency can be computed as

fc ≡∫ F (t0+Tobs)F (t0)

dff f

2|h(f)|2f∫ F (t0+Tobs)F (t0)

dff f

2|h(f)|2. (3.44)


Characteristic: h2c(F0) ≡∫ Fend

F0

dff f

2|h(f)|2 ; (Fend ≈ F0) ; ρ2opt(F0) ≈[hc(F0)hn(F0)

]2Effective: h2s(f) ≡ f2|h(f)|2 ; (Fend F0) ; ρ2opt(F0) ≈

∫∞F0

dff

[hs(f)hn(f)

]2Table 3.2: Summary of the characteristic and effective signals: definition, range of applicability andrelation to the optimal SNR. See discussion in Sec. 3.5 for further details; we recall that the effective (or

characteristic) noise amplitude is defined in (3.41): hn(f) ≡√

f4Sn(f).

Under these assumptions of narrow frequency band, the optimal SNR is simply the ratioof characteristic signal and noise,

ρ2opt =

h2c(fc)

h2n(fc)

. (when ∆(logF ) 1) (3.45)

In conclusion, effective and characteristic signals and noise consist in a more useful wayto (graphically) represent amplitudes and noise levels, respectively; first, because they areadimensional and second, because their ratio is directly related with the observed optimalSNR.

• When we are considering signals whose frequency evolution during the observationaltime can be actually appreciated in a log-scale plot spanning several orders of mag-nitude (for instance, Fig. 3.3), then we shall plot the effective signal amplitude (3.40)together with the effective noise, h2

n (3.41) and their squared ratio will represent theintegrand [over the log-frequency] of ρ2

opt (3.42); the integral can be estimated as thearea below the curve, assuming the log f variable as the base.

• On the other hand, when the frequency evolution (during a given observational time)is too small to be appreciated in a ‘log f ’ plot, then we shall compute the SNR integralbeforehand (assuming the effective noise level to be constant during that interval),resulting into the characteristic signal amplitude (3.43); then, the optimal SNR willbe just the ratio between the characteristic signal and the effective noise (3.45).

When considering GW signals from CBCs, it turns out that the same gravitational signalcan be in either of the two regimes, depending on the observational time and, mainly, onthe stage of the evolution that is being observed. Moreover, given the “abrupt” transitionin the frequency evolution close to the merger (see, for instance, bottom-right panel ofFig. 3.2), the transition zone between the two regimes will be very narrow. Hence, we shallnormally be observing either [Table 3.2 provides a summary of this discussion]

1. inspiral, quasi-monochromatic signals with F0 ≡ F (t0) ≈ F (t0 + Tobs) ≈ fc and thatwill be represented by their characteristic signal, hc(F0); or

2. the late-inspiral and the whole merger and ringdown stages of the coalescence, inwhich case, the frequency can vary by several orders of magnitude and hence, theeffective signal, hs(f), is providing a better representation of the signal (in terms ofrelating it to the SNR). In this case, since the coalescences are observed until theend, the upper limit of integration in Eq. (3.42), F (t0 +Tobs) ≡ Fend, can be replacedby infinity.


In the intermediate (and narrow) zone where ∆(logF ) > 1 but not by too much [in CBCs,when t0 + Tobs ≈ tmerger and F (t0 + Tobs) → ∞], the two representations (effective andcharacteristic) will still be valid within their definitions, i.e. the (squared) ratio of effectivequantities as integrand and the (squared) ratio of characteristic amplitudes as integralexpressions of the SNR2. Of course, they are not going to numerically coincide, as theyrepresent different quantities7, besides that each of them will have its own limitations inthat zone: (i) for characteristic amplitudes, as we go to higher frequencies, hc(F0) willrepresent an integral between a wide interval [F0, Fend] and the factorization of noiselevel out of the integral as hn(F0) will start being no longer valid; and (ii) for effectiveamplitudes, as we go to lower frequency values, the upper limit of the integral will not beinfinity anymore, and ∆(log f) will start being sufficiently small to make computation ofthe (now, very narrow) area below the curve less intuitive. The point where hc = hs, despitenot having any special meaning, will be in this intermediate zone and it will represent thefrequency F0 value where the ‘effective’ ∆ logF = 1, i.e. Fend/F0 ∼ e.

3.5.1 Analytical calculations within the Newtonian approximation

Before analyzing the effective and characteristic amplitudes obtained from the full EOBwaveforms already plotted in Figs. 3.1, 3.2 and 3.3, it is very interesting to start analyzingwhat kind of behavior do we expect. This can be easily (and analytically) done by justtaking the solution of the CBC dynamics to the leading Newtonian order [3]. This is,

|h(f)| =

√5π

24

ν1/2M2

Deff(πfM)−7/6 ≡ |A|f−7/6 ; (3.46)

F (t) =θ3

8πM, where θ =

[ ν

5M(tref − t)

]−1/8; (3.47)

t(F ) = tref −5M

256νv8, where v = (πFM)1/3 . (3.48)

In the previous equations, F is the GW frequency, M the total mass, ν = m1m2M2 the

symmetric mass ratio and tref represents the (reference) time such that the PN GW fre-quency diverges; of course this divergence does not occur when one considers full Einstein’sequations, hence the PN validity range always ends well before tref .

Effective and characteristic amplitudes

The effective amplitude signal, hs, is simply |h(f)| times f , so

h2s(f) = f2|h(f)|2 = f2|A|2f−7/3 = |A|2f−1/3 . (Newtonian) (3.49)

The computation of the characteristic signal, hc, actually involves an integral (3.43). Weshall consider two limit cases. On the one hand, we assume that we are close enough tothe merger so that Fend ≡ F (t0 + Tobs) > Fmerg and it can be considered as infinite,

h2c(F0) =

∫ ∞F0

df

f|A|2f−1/3 = 3|A|2F−1/3

0 = 3h2s(F0) . (Newtonian; Fend →∞)

(3.50)

7In fact, we shall see that in the Newtonian limit and assuming F (t0 + Tobs) → ∞, the two signalamplitudes are related as h2

c = 3h2s; see Eq. (3.50).


On the other hand, we can consider a system in the inspiral stage of the evolution, withsmall frequency variations, ∆(logF ) 1,

h2c(F0) =

∫ F (t0+Tobs)

F (t0)

df

f|A|2f−1/3

= 3|A|2[F (t0)−1/3 − F (t0 + Tobs)

−1/3]

= 3|A|2(8πM)1/3( ν

5M

)1/8(tref − t0)1/8

[1−

(1− Tobs

tref − t0

)1/8]

≈ 3|A|2(8πM)1/3( ν

5M

)1/8(tref − t0)1/8 1

8

Tobs

tref − t0= [· · · ] =

4ν2

D2eff

MTobs

π(πF0M)7/3 ;

finally,

hc(F0) =2ν

Deff

(MTobs

π

)1/2

(πF0M)7/6 . (Newtonian; ∆(logF ) 1) (3.51)

To summarize, the effective signal, hs(F0)nwt∝ F

−1/60 , provides intuitive information in

cases where there is a significant frequency change (in CBCs, when the observational timeincludes the merger and ringdown, Fend > Fmerg), whereas the characteristic signal am-

plitude, hc(F0)nwt∝ F

7/60 , shall be used to represent quasi-monochromatic signals in the

(deep) inspiral regime. In the transition zone, where F (t0 + Tobs) ≈ Fmerg, both represen-tations [with their respective meanings] will be valid and indeed, in Eq. (3.50) we see howh2c will end up being 3h2

s.

Transition and Last-Stable-Orbit zones

Given a certain observational time, Tobs, we shall start observing the merger stage of theevolution when F (t0 +Tobs) = Fmerg. The F0 ≡ F (t0) value associated with this condition,let us call it F0, will define the “transition zone” between considering quasi-monochromaticsignals [and therefore, using hc] to considering observations up to the end of the ring-down[and using hs]. We can compute these F0 values within the Newtonian approximation byusing Eq. (3.47) and assuming the PN merge to occur at tref ,

F0 ≡ F (tref − Tobs) =1

8πM

( ν

5MTobs

)−3/8=

1

8πM5/8

(5

ν Tobs

)3/8

. (3.52)

The effective amplitude associated to these F0 values, will be

hs(F0) =5

32π2Deff

1√6Tobs

F−3/20 . (Newtonian) (3.53)

Notice that all the dependency on M and ν remains implicit within F0, so we shall begetting the same ‘transition curves’ for all (ν,M) values (see Fig. 3.4).


Also, we can obtain the effective amplitude associated to the (test-mass particle) LSO,which is given by vlso = 1/

√6 or, in other words, Flso = 1/(63/2πM),

hs(Flso) =

√5ν

π3

1

63/4

1

12DeffF−1lso . (Newtonian) (3.54)

The previous expression has already been written explicitly in terms of Flso, but there stillremains a dependency on ν that will give different ‘LSO curves’ depending on the massratio. We also considered the possibility of using the mass-ratio-dependent expression forthe LSO given in Ref. [1], but the results we were obtaining were too complicated.

3.5.2 Plotting the effective/characteristic amplitudes

After defining which are the most appropriate quantities to intuitively represent signaland noise amplitudes, let us now take the frequency-domain amplitude of the gravitationalwaveform plotted in Fig. 3.3 [corresponding to an EOB simulation of an equal-mass, non-spinning BBH system including all stages of the evolution], and compute its associatedeffective and characteristic amplitudes. Results considering a fiducial effective distance ofDeff = 1 Gpc, an observational time of Tobs = 1 year and several total mass values, areplotted in Fig. 3.4.

In particular, we consider twelve different total mass values, M/M = 3, 15, 50, 200, 1000,104, 5× 104, 3× 105, 2× 106, 107, 5× 107, 3× 108; which basically respond to a log-scaleuniform grid from stellar-mass systems up to supermassive ones. However, there are someof the intermediate values that correspond to systems that have never been observed inthe Universe: from stellar evolution we know that BHs up to 20M [94] can exist and also,there are strong evidences to accept the presence of SMBHs in the center of every Galaxywith individual masses above ∼ 104M. Hence, for equal-mass CBCs there is a total massrange M ∈ (40M, 2× 104M), where it is unknown whether these systems can actuallyexist; we denote them as lighter blue curves in Fig. 3.4.

The evolution of the amplitude curves with the total mass of the system M , that can beobserved in Fig. 3.4, can be easily understood as a combination of two effects: on one hand,the signal’s amplitude increases with M , as |h(f)| ∝ M5/6 [see Eq. (3.46)]; on the otherhand, the amplitude curves shift to lower frequencies as M increases, since the solution toEinstein’s equations is generally obtained in terms of the adimensional frequency, fM .

The orange thick lines plotted in the same figure represent the transition zone from whereFend = F (t0 + Tobs) > Fmerg, i.e. if the observation starts at some point in the right handside of the orange line, it means that one shall observe the whole coalescing process upto merger within Tobs. The green thick line, on the other hand, represents the mergerposition (defined as the instant where the time-domain amplitude becomes maximum).Both loci have been numerically computed using the EOB waveform, but we observethat their general behavior is consistent with the analytical calculations done within theNewtonian approximation in the previous section: Eqs. (3.53) and (3.54), respectively.Also, the general trends of the effective and characteristic amplitudes, hs and hc, duringthe inspiral stage of the evolution are in concordance with the Newtonian predictions,Eq. (3.49) and (3.51), respectively. In particular, we recall that the effective amplitude,hs, consists simply in an adimensionalization of the original amplitude and it still decreases


10−4

10−2

100

102

104

10−23

10−22

10−21

10−20

10−19

10−18

10−17

10−16

10−15

f (Hz)

Effective o

r chara

cte

ristic a

mplit

udes

iLIGO

aLIGO

LISA

∝ f −3/2

∝ f −1

h c ∝

f7/

6

hs ∝ f−1/6

3Ms

15Ms

50Ms

200Ms

1000Ms

104Ms

5x104Ms

2x106Ms

107Ms

5x107Ms

x10

xe

Figure 3.4: Summary plot of the detectability of (equal-mass) CBCs (of any total mass) with current andfuture interferometric GW detectors. The thick green line represents the locus where the merger (defined asthe instant where the time-domain amplitude is maximum) occurs, for different total mass values; whereasthe thick orange line represents the frequency and amplitude values such that F (t0 + Tobs) = Fmerg,assuming Tobs = 1 yr; in other words, it represents the transition zone of usefulness of the characteristicand effective signals definitions (see discussion and further details in the main text). In particular, theblue lines (both light and dark blue) represent characteristic signal amplitudes, hc, in the left hand sideof the thick orange line and effective amplitudes, hs, in the right hand side; always considering equal-mass systems at a fiducial effective distance of Deff = 1 Gpc and twelve different total mass values,M/M = 3, 15, 50, 200, 1000, 104, 5× 104, 3× 105, 2× 106, 107, 5× 107, 3× 108. The darkness level of theblue lines gives an idea of the odds of finding this kind of systems in the Universe. We also overprint theeffective noise amplitude, hn, of both ground- and space-based interferometric GW detectors. Finally, thegreen segments on the top-right corner give an idea of what a factor 10 and e are in the ‘y’ and ‘x’ axis,respectively. [Plot: own production, using an EOB simulation courtesy of A. Nagar and T. Damour]


10−4

10−2

100

102

104

10−23

10−22

10−21

10−20

10−19

10−18

10−17

10−16

10−15

f (Hz)

Effective o

r chara

cte

ristic a

mplit

udes

iLIGO

aLIGO

LISA

x10

x e

10−4

10−2

100

102

104

10−23

10−22

10−21

10−20

10−19

10−18

10−17

10−16

10−15

f (Hz)

Effective o

r chara

cte

ristic a

mplit

udes

iLIGO

aLIGO

LISA

x10

x e

10−4

10−2

100

102

104

10−23

10−22

10−21

10−20

10−19

10−18

10−17

10−16

10−15

f (Hz)

Effective o

r chara

cte

ristic a

mplit

udes

iLIGO

aLIGO

LISA

x10

x e

(a) Deff = 0.15 Mpc (b) Deff = 30 Mpc (c) Deff = 6.3 Gpc; (z = 1)[Galactic realm] [Virgo supercluster] [‘cosmological’ distances]

Figure 3.5: Similar plots to Fig. 3.4, but considering three particular effective distance values, Deff , thatrepresent the typical size of astronomical structures that contain the Solar System within. We use threedifferent darkness levels for the (blue) signal’s amplitude curves, in order to represent (in an approximateand qualitative way) the odds of having CBCs of such total mass, M , within the different volumes. Weconsider the same twelve M values as in Fig. 3.4. [Plots: own production, using an EOB simulation courtesyof A. Nagar and T. Damour]

with frequency; however, the characteristic amplitude hc represents already an integratedquantity over a fixed Tobs. So, despite the fact that the original amplitude in the frequency-domain was higher at lower frequencies, the frequency variation (for a fixed Tobs) becomessmaller as one moves to smaller frequency values, making the integral expression, hc, todecrease as we move to lower frequencies.

As we have seen in Sec. 3.4, the emission frequency of a CBC is monotonically increasing,but not at a constant rate, very slowly in the early stages of the evolution and much fasteras one gets close to the merger. In terms of the amplitude curves represented in Fig. 3.4, thisone-to-one relation between frequency and time means that any coalescing binary system(with a fixed total mass8, M) moves along a single line during its lifetime: the higherthe frequency is, the closer to the merger. Furthermore, the increase on the frequencyderivative as the system approaches the merger, translates into a changing velocity ofhow the system moves along the amplitude curves. For instance, by definition any systemspends Tobs = 1 yr to go from the orange line up to merger, but during this very same time,when the system is at some point at the l.h.s. of the orange line, one can not even noticethe frequency variation. Indeed, at the l.h.s. of the orange line we can find systems that arefrom millions of years, up to one year, prior their merger; whereas at the r.h.s. all systemshave less than one year left of life time. This is very important in order to understandthe population distribution of sources along the amplitude curves: if we assume a uniformdistribution of CBC ages, we shall expect millions of times more sources at the l.h.s. ofthe orange thick line than at the r.h.s.

In the same plot in Fig. 3.4, we also show the effective (or characteristic) noise amplitudesof both ground and space based GW interferometric detectors, in particular initial LIGO[95], advanced LIGO [95] and LISA [96].

With all the information presented in a figure like Fig. 3.4 and undestanding the meaning ofeffective and characteristic amplitudes (see Tab. 3.2 for a summary), one can easily obtain

8If we assume that the emitted energy is small compared with the total mass of the system; which isnot true in reality.


very useful information about (i) the systems that will be observable in each detector, (ii)which parts of the coalescing process will be visible and (iii) the typical SNR values ata given effective distance. Given the inverse proportional relation between h(f) and Deff ,an increase of an order of magnitude in the effective luminosity distance would representa decrease on both effective and characteristic signals by the same factor. In particular,assuming a fiducial effective distance of 1 Gpc as it is done in Fig. 3.4 [which representsa volume ∼ 104 larger than the size of the Virgo supercluster]; we see that Adv. LIGOcould observe the late inspiral of a system with M = 15M and the merger would becomevisible for higher mass systems, such as M = 50M and M = 200M. LISA, on the otherhand, would be able to observe the quasi-monochromatic signals from the deep inspiralregime of systems with M ∈ (200M, 107M) and the actual mergers of systems withM > 5× 104. Moreover, the expected SNRs observed by LISA would be much higher thatwhat we expect in ground-based detectors.

The effective distance considered in Fig. 3.4 is arbitrarily chosen. In fact, we think thatsomehow one can obtain even more useful information when one considers astronomicallymotivated Deff values and also total mass ranges according to the expected population.Indeed, this study is what it is done in Fig. 3.5, considering three different scenarios: (a)the Galactic realm [Deff = 0.15 Mpc], (b) the Local (Virgo) supercluster [Deff = 30 Mpc]and (c) ‘cosmological’ distances [Deff = 6.3 Gpc]. With these plots and some astronomicalarguments, we shall be able to predict which are the gravitational signals generated from(similar-mass) CBCs that are expected to be observed in the ground-based or space-basedfrequency bands, and also their properties and origin.

• Galactic realm. Massive BHs are located at the center of galaxies [97–99] (for in-stance, we know that there is a 4.5 × 106M SMBH at the center of our galaxy,see e.g. [100]), so massive events will be related to galaxy mergers [101] and we donot expect such events at distances of the order of the Milky Way scale and neitherin the Local supercluster [thus, in Fig. 3.5a-b we shall represent these systems withthe lightest blue color]. What we do expect is to have an important population ofstellar-mass BHs generated as the end product of massive stars evolutions. An smallfraction of these BHs can be in binary systems and therefore emit GWs.

In principle, we see that the same systems could be observed by LISA in their deep-inspiral stage (with SNRs from tens up to hundreds) and also by LIGO when theyare close to the merger (SNR ∼ 100 − 1000). The coalescence rate of BBH systemsis 0.4 Myr−1 per Milky Way Equivalent Galaxy [95], so we do not expect to observeany BBH merger from our galaxy; however LISA will observe such systems (indeed,millions of them) in an early stage of their evolution, where they remain for a verylong time and therefore, they have a much greater population.

Thus, the only (similar-mass) CBCs that we expect to observe from our galaxy arestellar-mass objects in their deep inspiral stage of the evolution, emitting quasi-monochromatic signals; we shall refer to them as galactic binaries. Despite beinga CBC, they are too far away from merger that their gravitational waveform forTobs = 1 yr, can be modeled as a monochromatic signal plus an spin-up parameterrepresenting the first frequency derivative. Galactic binaries are by far the most nu-merous among LISA sources, and indeed most of them will be unresolvable becomingan extra source of noise, so-called confusion noise. Trying to resolve them representsone of the major challenges for LISA data analysis, not only in order to obtain indi-vidual information from them, but also to remove them from the data and therefore


reduce the confusion noise that also affects other LISA sources. We expect to be ableto identify several thousands of such systems during LISA’s mission life time.

In this thesis, we have developed MCMC-based methods to search for a fixed numberof galactic binary signals buried in LISA noise, and also we have participated in theMock LISA Data Challenges (MLDC) [102, 103]. (see Chapters 6 and 7 for furtherdetails).

• Local (Virgo) supercluster. We don’t expect to observe any SMBH binary mergerevent coming from the Local supercluster neither, but having increased the volumeby a factor 8 × 106 compared to the Galactic realm increases the chances to ob-serve stellar-mass BBH mergers. Indeed, the expected CBC signals to be observedby ground-based detectors will be located within the Local supercluster and evena bit beyond. We observe in Fig. 3.5b how the expected SNR is still high enoughto make observations and the CBC rates start being close to 1 yr−1. In particular,recently in Ref. [95] the LIGO and Virgo Collaborations have revisited the issueof predictions for the rates of CBC observable by ground-based detectors, assuminghorizon distances of 161 Mpc/2187 Mpc for BH-BH inspirals in the Initial/AdvancedLIGO-Virgo networks, and obtaining detection rates of 0.007 yr−1/20 yr−1, respec-tively. Notice that for very low mass systems (M . 10M) only the late inspiral partof the waveform will be observable, but as we increase a little bit the total mass, themerger part becomes relevant. Indeed, this has been the main motivation for scien-tists working on Numerical and Analytical Relativity problems over the last yearsin order to obtain an accurate description of the dynamics of CBC systems aroundthe merger. Indeed, part of the work done during this thesis has been devoted tostudy the performance (in terms of accuracy and effectualness) of the (fast) closed-form, frequency domain waveform models in comparison to the slower, but moreaccurate, EOB templates, see Chapter 8; with this study we get to know the rangeof applicability of the fastest waveform models, either for detection or measurementpurposes.

Part of the extragalactic stellar-mass compact systems in the deep-inspiral regime arestill visible in the LISA band and indeed, there is also an extragalactic component ofconfusion noise. However, looking at Fig. 3.5b, at these distances, one has to considersomehow high M values in order to obtain ‘observable’ (> 10) SNR values, and theLISA noise curve plotted here does not contain the contribution from confusion noise.In conclusion, there is an extragalactic component to the confusion noise, but we donot expect to individually resolve any extragalactic stellar-mass binary.

• ‘Cosmological distances’. Assuming a spatially flat Friedman-Lemaıtre-Robertson-Walker (FLRW) universe with H0 = 75 km s−1Mpc−1, ΩM = 0.27, ΩDE = 0.73,and w = −1, a luminosity distance of Deff = 6.3 Gpc corresponds to a redshiftz = 1; and we could even consider longer distances, such as Deff = 15 Gpc (z = 2),Deff = 45 Gpc (z = 5) or Deff = 100 Gpc (z = 10). In Fig. 3.5c, we see thatLISA will be able to observe massive and supermassive BBHs coalescences from anymass within M ∈ (104M, 108M), at almost any distance, with SNRs of hundredsand even thousands. In most of the systems, the merger part can have a significantcontribution to the SNR (if Tobs is long enough), so it will be also important to includeinformation from either EOB or NR simulations. Given the fact that the detectionproblem of SMBHs signals with LISA is relatively easy [because of the high SNRvalues]; during the past years, data analysts have focussed on parameter estimation


studies. In particular, one of the main topics of this thesis (Chapter 4) have beento study the impact on parameter estimation of adding the higher harmonics ofthe orbital frequency to the inspiral PN signal, finding significant improvements onrelevant parameters.

With long observational times (∼ months), one of the parameters that LISA candirectly measure is the luminosity distance. This is not normally the case with other(electromagnetic) astronomical measurements, where one obtains accurate measure-ments of the redshift by looking at the emitted spectra, but it is very hard to directlymeasure luminosity distances. This LISA’s feature, combined with the fact that itwill observe SMBHs coalescence events from anywhere in the Universe, makes thespaced-based detector a powerful tool to do cosmography. In particular, during thethesis, we have studied LISA’s performance in measuring the dark energy equationof state from a single SMBHs inspiral event at z ≤ 1 (Chapter 5).

Advanced ground-based detectors and, specially, the planned third generation ofinterferometers, might also observe some merger events at ‘cosmological distances’,Deff ∼ Gpc. We can see in Fig. 3.5c how the late inspiral, merger and ringdownparts of the waveform will have a key role in these observations; thus, the better ourunderstanding of the signals emitted in that regime is (including higher harmonics,spins, precession, etc.), the better the cosmological outstanding of the advancedground-based detectors will be.

In conclusion, LISA will observe thousands of stellar-mass BBHs coalescences from ourgalaxy when they are still in the deep-inspiral stage of the evolution (galactic binaries),i.e. emitting quasi-monochromatic signals; also it will observe SMBHs inspirals and merg-ers from almost any place of the Universe [up to z = 10− 15]. Ground-based detectors, onthe other hand, will observe the late inspiral, merger and ringdown phases of stellar-massand up to several hundreds of solar masses CBCs produced within ∼ 150 Mpc (2000 Mpc),when one considers Initial (Advanced) detectors; this represents a volume ∼ 100 (105)times larger than the Virgo supercluster.

References 81

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Part II

Original scientific results

85

Chapter4Impact of higher harmonics onparameter estimation of SMBHsinspiral signals

The following published articles were included in this chapter of the thesis:

• M. Trias and A.M. Sintes, LISA observations of supermassive black holes: parameterestimation using full PN inspiral waveforms, Physical Review D 77, 024030 (2008)[19 pages]

• M. Trias and A.M. Sintes, LISA parameter estimation of supermassive black holes,Classical and Quantum Gravity 25, 184032 (2008) [9 pages]

• K.G. Arun et al. (including M. Trias), Massive Black Hole Binary Inspirals: Resultsfrom the LISA Parameter Estimation Taskforce, Classical and Quantum Gravity 26,094027 (2009) [14 pages]

87

Chapter5Measuring the dark energyequation of state with LISA

The following published article was included in this chapter of the thesis:

• C. Van Den Broeck, M. Trias, B.S. Sathyaprakash and A.M. Sintes, Weak lensingeffects in the measurement of the dark energy EOS with LISA, Physical Review D81, 124031 (2010) [15 pages]

133

Chapter6Searching for galactic binarysystems with LISA using MCMC

The following published articles were included in this chapter of the thesis:

• M. Trias, A. Vecchio and J. Veitch, MCMC searches for galactic binaries in MLDC1B data sets, Classical and Quantum Gravity 25, 184028 (2008) [10 pages]

• M. Trias, A. Vecchio and J. Veitch, Studying stellar binary systems with LISA usingDR-MCMC methods, Classical and Quantum Gravity 26, 204024 (2009) [11 pages]

151

Chapter7Delayed Rejection Markov chainMonte Carlo

The following article was included in this chapter of the thesis:

• M. Trias, A. Vecchio and J. Veitch, DR schemes for efficient MCMC sampling ofmultimodal distributions, arXiv: 0904.2207 [31 pages]

175


Addendum to Chapter 7. Toy examples to illustrate andquantify the efficiency of DR MCMC chains

The benefit of DR is that it increases the efficiency of sampling by delaying a possible arejection, at the same time that it allows us to propose a new candidate that can be drawnusing the information of the past stages. Indeed, Mira et al. [1, 2] demonstrate from aformal point of view that DR increases the transition probability of the states of the chainwithin the different parts of the parameter space, improving the mixing of the chain andleading to a reduction [3] of the asymptotic variance of any estimator, say λ.

The purpose of this addendum is to ‘illustrate’ this efficiency increase in some toy examples,by estimating and comparing the variance of chains generated with and without DR. Weare fully aware that all the results that we shall obtain in this addendum are merely testsof validity, applied to simple examples, of the general result formally demonstrated by Miraet al. [1, 2] and therefore the conclusions we shall obtain here, are not new. However, wethink that it may be useful to observe the benefits of DR in some actual examples, at thesame time that we obtain a quantitative measurement of this efficiency increase in termsof the variances of the resulting chains.

Intuitively, it may be difficult to see that DR actually increases the efficiency of the chainat a given computational cost1 because, on the one hand, it reduces the number of repeatedelements of a chain by delaying a possible rejection; but on the other hand, the numberof elements of the resulting chain (for a fixed computational cost, i.e. number of newproposed states2) will be smaller. Since the variance of the chain involves the ratio betweenthe correlation function and the number of elements of the chain [see Eq. (7.8) below], itis not clear which of the two effects shall dominate, or whether this can depend on theparticular problem at hand. Thus, the question that we want to answer in this addendumis: “Is there a benefit (in terms of reducing the variance of the estimate) from removingsome repeated elements of a chain?”.

Preliminary considerations

Suppose we wish to estimate the expectation

I ≡ 〈f〉π =s∑i=0

f(i)πi , (7.1)

where π = (π0, π1, . . . , πs) is a positive probability distribution, i.e. πi > 0 for all i, andf(·) is a non-constant function defined on the states 0, 1, . . . , s of an irreducible Markovchain determined by the transition matrix P = pij. If P is chosen so that π is its uniquestationary distribution, then after simulating the Markov chain for times t = 1, . . . , N , anestimate of the expectation I is given by

I =N∑t=1

fλ(t)/N , (7.2)

1We assume that evaluating the likelihood function is the most demanding calculation in terms of com-putational time, and therefore we shall estimate ‘computational costs’ directly as the number of likelihoodevaluations.

2Each new proposed state requires a likelihood evaluation.

210 Chapter 7: Delayed Rejection Markov chain Monte Carlo

where λ(t) denotes the state occupied by the chain at time t.

The performance in the MCMC sampling can be measured through two quantities: (i)the bias of the estimate I, defined as limN→∞

[E(I)− I

], represents the accuracy of the

sampling; and (ii) the variance of the estimate, which represents its precision. As Peskun[3] points out, since var(I) is O(N−1) and the bias squared is O(N−2), for an appropriatedistribution of the initial state λ(0), the bias has a negligible effect (at least, it decreasesfaster) on the accuracy of the estimate I. Moreover, we can highly reduce the bias byneglecting the points from the ‘burn-in’ period. We shall thus confine our discussions tothe precision (variance), rather than the accuracy (bias), of the estimate I.

In particular, the variance of the estimate I can be computed in terms of the autocorrela-tion function of the Markov chain [4–6] as

var(I) =1

N2

N∑r,s=1

Cff (r − s) , (7.3)

where Cff (t) is the (unnormalized) autocorrelation function of the function fλ at lag t,defined as

Cff (t) ≡ 〈fsfs+t〉 − I2 = 〈fλ(s)fλ(s+ t)〉 − I2 . (7.4)

The normalized autocorrelation function is then

ρff (t) ≡ Cff (t)/Cff (0) . (7.5)

Typically, ρff (t) decays exponentially for large t; this defines the exponential autocorrela-tion time, τexp, [4]

ρff (t) ∼ e−|t|/τexp . (7.6)

When the number of elements of the Markov chain, N , is large enough; the varianceof the estimate I given by equation (7.3) can be rewritten in terms of the integratedautocorrelation time [4],

τint ≡1

2

N∑n=−N

ρff (n) =1

2+

N∑n=1

ρff (n) , (7.7)

as

var(I) ' 2 τint Cff (0)

N. (7.8)

The factor of 12 in the definition of the integrated autocorrelation time (7.7) is purely a

matter of convention; it is inserted so that τint ≈ τexp for τ 1.

In what follows, we shall consider several toy examples in which we shall assume to have ahypothetical Markov chain with a certain number of repeated elements (namely, chain A)that could be removed by using DR, resulting into a shorter, but less correlated, chain (letus call it chain B). Given the simplicity (and periodicity) of the Markov chains consideredin these toy examples, we shall be able to compute, in a theoretical way and using Eqs. (7.7)and (7.8), the variance of an arbitrary estimate I from the chain B, varB(I), in terms of thevariance of the same estimate from the chain A, varA(I). Despite being just toy examples,we aim to provide a quantitative illustration of the benefits of using DR.


Chain A (duplicated elements): ① ① ② ② ③ ③ ④ ④ ⑤ ⑤ ⑥ ⑥ ⑦ ⑦ ⑧ ⑧ ⑨ ⑨ ⑩ ⑩ …

Chain B: ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ …

Figure 7.1: Case I. Example of two Markov chains, one of them with all its elements duplicated.

Case I: Duplicate all elements of a chain

We shall start considering a very simple scenario, which consists in assuming that only oneevery second proposed state is accepted, resulting into a Markov chain with all its elementsduplicated, as every time one proposal is rejected the chain remains at the original state.In this case, a two-stages DR algorithm could be used in order to remove all the repeatedelements of the chain. Fig. 7.1 provides an schematic representation of the two chains,using repeated circled numbers to denote repeated elements of the chain.

Without loss of generality, we shall assume a zero-mean estimator, i.e. I = 0, so that theautocorrelation function at lag t, Eq. (7.4), is simply Cff (t) = 〈fsfs+t〉 = 1

N−t∑N−t

s=1 fsfs+t.

From Fig. 7.1, and assuming that the chains only have the finite number of elementsexplicitly shown in the figure, we can explicitly compute the autocorrelation function ofthe two different chains and look for a relation between the different values.

CB(0) =1

10

(12 + 22 + 32 + 42 + · · ·+ 102

)(7.9)

CA(0) =1

20

(2 12 + 2 22 + 2 32 + 2 42 + · · ·+ 2 102

)= CB(0)

CB(1) =1

9( 1 2 + 2 3 + 3 4 + 4 5 + · · ·+ 9 10) (7.10)

CA(1) =1

19

(12 + 1 2 + 22 + 2 3 + · · ·+ 9 10 + 102

)=

1

19[10CB(0) + 9CB(1)]

CB(2) =1

8( 1 3 + 2 4 + 3 5 + 4 6 + · · ·+ 8 10) (7.11)

CA(2) =1

18( 1 2 + 1 2 + 2 3 + 2 3 + · · ·+ 9 10 + 9 10) = CB(1)

CB(3) =1

7( 1 4 + 2 5 + 3 6 + 4 7 + · · ·+ 7 10) (7.12)

CA(3) =1

17( 1 2 + 1 3 + 2 3 + 2 4 + · · ·+ 8 10 + 9 10) =

1

17[9CB(1) + 8CB(2)]

CB(4) =1

6( 1 5 + 2 6 + 3 7 + 4 8 + · · ·+ 6 10) (7.13)

CA(4) =1

16( 1 3 + 1 3 + 2 4 + 2 4 + · · ·+ 8 10 + 8 10) = CB(2)

CB(5) =1

5( 1 6 + 2 7 + 3 8 + 4 9 + 5 10) (7.14)

CA(5) =1

15( 1 3 + 1 4 + 2 4 + 2 5 + · · ·+ 7 10 + 8 10) =

1

15[8CB(2) + 7CB(3)]

· · ·


Chain A (duplicated elements): ① ② ② ③ ④ ④ ⑤ ⑥ ⑥ ⑦ ⑧ ⑧ ⑨ ⑩ ⑩ …

Chain B: ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ …

Figure 7.2: Case II. Example of two Markov chains, one of them with its even elements duplicated.

From these explicit expressions, one can infer the general relation between CA(n) andCB(n) for all n:

CA(0) = CB(0) ,

CA(2n) = CB(n) , (7.15)

CA(2n+ 1) =1

2N − 2n− 1[(N − n)CB(n) + (N − n− 1)CB(n+ 1)] .

Finally, we use Eq. (7.7) to compute the integrated autocorrelation time, that is directlyrelated to the variance of the estimate.

τint, B =1

2+

1

CB(0)

N∑n=1

CB(n) ; (7.16)

τint, A =1

2+

1

CA(0)

2N∑n=1

CA(n)

=1

2+

1

CB(0)

(N∑n=1

CA(2n) +

N−1∑n=0

CA(2n+ 1)

)

=1

2+

1

CB(0)

(N∑n=1

CB(n) +N−1∑n=0

N − n2N − 2n− 1

CB(n) +N−1∑n=0

N − n− 1

2N − 2n− 1CB(n+ 1)

)

≈ 1

2+

1

CB(0)

(N∑n=1

CB(n) +1

2

N−1∑n=0

CB(n) +1

2

N−1∑n=0

CB(n+ 1)

)

≈ 1

2+

1

CB(0)

(N∑n=1

CB(n) +N∑n=1

CB(n) +1

2CB(0)

)= 2 τint, B ; (7.17)

where we only have assumed N →∞, and that CB(n)→ 0 as n→ N .

Thus, in this case we obtain that by duplicating N ; τ also doubles, keeping the variance(7.8) constant: varA(I) = varB(I). Hence, in this extremely simple example the use of DRdoes not really increase the efficiency of the Markov chain since the effect of reducing itscorrelations is compensated by the decrease on the number of elements of the chain (forthe same computational cost). We shall see in the following examples, that this is not thegeneral case and that normally, the use DR does increase the efficiency of the resultingchains.

Case II: Duplicate one element every second

In this case, we go one step further and consider a chain that only its even elements areduplicated (see Fig. 7.2). This case could exemplify a canonical situation where a proposalis accepted twice every three iterations. With this, the explicit autocorrelation functions


take the form,

CB(0) =1

10

(12 + 22 + 32 + 42 + · · ·+ 102

)(7.18)

CA(0) =1

15

(12 + 22 + 22 + 32 + 42 + 42 + · · ·+ 92 + 102 + 102

) avg.≈ CB(0)

CB(1) =1

9( 1 2 + 2 3 + 3 4 + 4 5 + · · ·+ 9 10) (7.19)

CA(1) =1

14

(1 2 + 22 + 2 3 + 3 4 + 42 + · · ·+ 9 10 + 102

) avg.≈ 1

14[9CB(1) + 5CB(0)]

CB(2) =1

8( 1 3 + 2 4 + 3 5 + 4 6 + · · ·+ 8 10) (7.20)

CA(2) =1

13( 1 2 + 2 3 + 2 4 + 3 4 + 4 5 + 4 6 + · · · ) avg.≈ 1

13[9CB(1) + 4CB(2)]

CB(3) =1

7( 1 4 + 2 5 + 3 6 + 4 7 + · · ·+ 7 10) (7.21)

CA(3) =1

12( 1 3 + 2 4 + 2 4 + 3 5 + 4 6 + 4 6 + · · · ) avg.≈ CB(2)

CB(4) =1

6( 1 5 + 2 6 + 3 7 + 4 8 + · · ·+ 6 10) (7.22)

CA(4) =1

11( 1 4 + 2 4 + 2 5 + 3 6 + · · ·+ 7 10 + 8 10)

avg.≈ 1

11[7CB(3) + 4CB(2)]

CB(5) =1

5( 1 6 + 2 7 + 3 8 + 4 9 + 5 10) (7.23)

CA(5) =1

10( 1 4 + 2 5 + 2 6 + 3 6 + · · · ) avg.≈ 1

10[7CB(3) + 3CB(4)]

CB(6) =1

4( 1 7 + 2 8 + 3 9 + 4 10) (7.24)

CA(6) =1

9( 1 5 + 2 6 + 2 6 + 3 7 + · · ·+ 6 10)

avg.≈ CB(4)

· · ·

Here, in order to obtain the relations after theavg.≈ symbol, we have used the fact that the

Markov chain samples an stationary distribution.

From these explicit expressions, one can infer the general relations between CA(n) andCB(n) for all n:

CA(0) = CB(0) ,

CA(3n) = CB(2n) , (7.25)

CA(3n+ 1) = 132N−3n−1

[(N − 2n− 1) CB(2n+ 1) +

(N2 − n

)CB(2n)

],

CA(3n+ 2) = 132N−3n−2

[(N − 2n− 1) CB(2n+ 1) +

(N2 − n− 1

)CB(2n+ 2)

].


Then, the general expressions for the integrated autocorrelation times will be given by

τint, B =1

2+

1

CB(0)

N∑n=1

CB(n) (7.26)

τint, A =1

2+

1

CA(0)

3N/2∑n=1

CA(n)

=1

2+

1

CB(0)

N/2∑n=1

CA(3n) +

N2 −1∑n=0

CA(3n+ 1) +

N2 −1∑n=0

CA(3n+ 2)

=

1

2+

1

CB(0)

N/2∑n=1

CB(2n) +

N2 −1∑n=0

[N − 2n− 132N − 3n− 1

CB(2n+ 1) +N2 − n

32N − 3n− 1

CB(2n)

+N − 2n− 132N − 3n− 2

CB(2n+ 1) +N2 − 2n− 1

32N − 3n− 2

CB(2n+ 2)

])

≈ 1

2+

1

CB(0)

N/2∑n=1

CB(2n) +1

3

N2 −1∑n=0

[2CB(2n+ 1) + CB(2n) + 2CB(2n+ 1) + CB(2n+ 2)]

=

1

2+

1

CB(0)

N/2∑n=1

CB(2n) +4

3

N2 −1∑n=0

CB(2n+ 1) +1

3CB(0) +

2

3

N2 −1∑n=0

CB(2n+ 2)

=

1

2+

1

CB(0)

4

3

N∑n=1

CB(n) +1

3

CB(0) +

N/2∑n=1

CB(2n)

=

4

3τint, B +

1

3

1

2+

N/2∑n=1

CB(2n)

CB(0)

(7.27)

Making use of the definition of the exponential autocorrelation time in Equation (7.6),we can rewrite τint, B and the sum that appears in the final result of (7.27), in a moreconvenient way [here we shall define τexp,B ≡ τ and assume N τ ]:

τint, B =1

2+

N∑n=1

CB(n)

CB(0)=

1

2+

N∑n=1

e−n/τ =1

2+

1

e1/τ − 1=

1

2coth

[1

2τ

], (7.28)

(·) in (7.27) =1

2+

N/2∑n=1

CB(2n)

CB(0)=

1

2+

N/2∑n=1

e−2n/τ =1

2+

1

e2/τ − 1=

1

2coth

[2

2τ

]. (7.29)

Finally, we can put together all the results from Eqs. (7.26)-(7.29) into the definition ofthe variance of the estimate I, Eq. (7.8), in order to obtain by how much the variance is


reduced because of removing the repeated elements of the chain [we recall that NA = 32NB],

varA(I)− varB(I)

varB(I)=

τint, A − NANB

τint, BNANB

τint, B

=

43 τintB + 1

3

(12 +

∑N/2n=1

CB(2n)CB(0)

)− 3

2τintB32 τintB

=2

9

(12 +

∑N/2n=1

CB(2n)CB(0)

)τint, B

− 1

9

=1

9

(2 coth

[22τ

]tanh

[12τ

]− 1)≥ 0 (7.30)

The main consequence of this result is that varB(I) ≤ varA(I) or, in other words, wehave just showed that DR always increases the efficiency of the resulting Markov chainin terms of reducing the variance of any estimate. Moreover, the previous result allowsus to obtain quantitative measures of such improvement; in particular, we would obtaina 2.4% reduction if τ = 1 ; 0.7% when τ = 2 ; 0.3% when τ = 3; . . . These are indeedmoderate improvements, but as we shall see below in Fig. 7.4, DR performance becomesimportant when the needed number of stages within a DR sub-chain is much greater thanthe autocorrelation time of the (main) chain.

Case III: Repeat M2 times one element every M1-th

Finally, let us consider a generalization of the two previous examples, where we shallassume a chain consisting in a set of M2 repeated elements every M1-th, M1 and M2 beingintegers. We see this toy model as a basic representation of a situation in which a Markovchain is generated from two different proposals, P1 and P2, and one of them, say P2, hasa much lower acceptance probability (see Fig. 7.3). Then, as far as one is proposing fromP1, a chain is being built with a given autocorrelation time τ ; and once every M1 of theseproposals, one attempts P2 until it is accepted, which we assume that it is M2 times.The resulting chain (A) will contain M2 completely correlated (repeated) elements everyM1 +M2 iterations, whose efficiency could be improved by grouping all these M2 elementsinto a single one, using the DR algorithm (chain B in Fig. 7.3).

This particular example tries to emulate, in a sense, the situation described in the articlepreviously included in the current chapter; where P1 would be the unimodal Gaussianproposals that explore the neighborhood of the original position, and P2 would representthe trimodal distributions attempting big jumps in the parameter space. When the likeli-hood function is characterized by several isolated maxima, the acceptance probability ofP1 will be much higher than P2. Another situation that could be well exemplified by thecurrent toy model is a case where we would use delayed rejection (DR) in combinationwith RJMCMC in order to delay the possible rejection of transdimensional transitions;in this case, P1 would be the proposals exploring the parameter space within the samemodel, and P2 would represent the transdimensional proposals.

Following a similar approach than in the previous cases, we can compute the generalexpressions that relate the autocorrelation functions of the two chains schematized in


P1

P1

(M2-1) P

2

P1

P1

P1

P1

P2

P2

P2

P1

P1

P1

P1

P1

P2

P2

P2

P1

P1

P1

P1

P1

P2

P1

P1

P1

P1

P1

P2

P1

(M2-1) P

2

Chain A (no DR): ① ② ③ ④ ⑤ ⑥ ⑥ ⑥ ⑦ ⑧ ⑨ ⑩ ⑪ ⑫ ⑫ ⑫ ⑬ …

Chain B (with DR): ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ⑪ ⑫ ⑬ …

Figure 7.3: Case III. Example of the Markov chains that would be generated in the theoretical case inwhich every M1 elements drawn from a certain proposal, P1, M2 transitions from a second proposal, P2,are attempted but accepted with lower probability. The Delayed Rejection algorithm attempts the samenumber (M2) of transitions drawn from P2 as the standard MCMC, but it only stores the one that isaccepted. Different circled numbers represent possible different elements of the chain, whereas the repeatedones explicitly denote the rejected transitions drawn from P2. In this example we are considering M1 = 5and M2 = 3.

(a) (∆var/var)× 100 (b) M1 that maximizes (∆var/var)

Figure 7.4: Maximization of Eq. (7.31) over M1 assuming different values for M2 and τ . For a givenM2, τ values, contour plot ‘(a)’ shows the (maximized) percentage reduction of the variance when therepeated elements of the chain are removed (i.e. when DR is used); whereas ‘(b)’ plots the M1 values thatmaximized this variance reduction.

Fig. 7.3, A and B; then we use Eq. (7.7) to compute the integrated autocorrelation time foreach of the chains; and, finally, one makes use of Eq. (7.8) to compare the two variances.The final result that it is obtained after adopting similar approximations as the onesdescribed in the previous cases is:

varA(I)− varB(I)

varB(I)=

(M2 − 1

M1 +M2

)2 [(M1 + 1) coth

(M1+1

2 τ

)tanh

(1

2 τ

)− 1]≥ 0 , (7.31)

where we have used the definition of the exponential autocorrelation time, τexp,B ≡ τ ,given in (7.6). In Fig. 7.4 we shall denote the quantity defined here, in Eq. (7.31), as(∆var/var).

Notice that, although ‘a priori’ and intuitively it could be difficult to see which of theeffects related to the efficiency of the chain (to reduce the autocorrelation by removingrepeated elements; or to reduce the length of the chain for a fixed computational cost;see discussion at the beginning of this addendum) would dominate; the result obtained inEquation (7.31) shows that, for a fixed computational time, we always obtain a benefit fromremoving repeated elements of a chain in terms of reducing the variance of any estimate.


It is clear that the current toy model we are considering here is just a canonical represen-tation of the chains usually found in actual applications of DR, but at least it providesan approximated, and quantitative, result that supports the formal proofs provided byMira et al. [1, 2]. Besides obtaining that varB(I) ≤ varA(I), Eq. (7.31) also can be used toobtain some quantitative information as a function of M1,M2, τ. In particular, we shallconsider M2, τ to be given by the problem at hand [M2 as a typical value for the in-verse of the acceptance probability of the P2 proposal, and τ as the typical autocorrelationtime of the elements generated from P1], thus we are studying the variance reduction as afunction of M1 [we recall that this parameter represents how often a set of P2 proposals isattempted]. It turns out that Eq. (7.31) always shows a single local maximum as a functionof M1 (keeping M2, τ fixed); and, indeed, Fig. 7.4 plots the result of this maximizationprocess [(a) the maximum values of the percentage variance reduction, and (b) the M1

values that maximized (∆var/var)] as a function of M2, τ. We observe that the mostimportant variance reductions are obtained when M2 is large and τ small, i.e. when theacceptance probability of P2 is much smaller than P1’s. In these cases, we observe that thereductions can be & 100%.


References

[1] A. Mira, Metron LIX (3-4), 231-241 (2001).

[2] L. Tierney and A. Mira, Statistics in Medicine 18 (17-18), 2507-2515 (1999).

[3] P. H. Peskun, Biometrika 60 607 (1973).

[4] A. D. Sokal, Monte Carlo methods in statistical mechanics: foundations and new algorithms,lectures given at the Cours de Troisieme Cycle de la Physique en Suisse Romande, Lausanne,Switzerland (1989).

[5] W. Gilks, S. Richardson and D. Spiegelhalter, Markov Chain Monte Carlo in practice, (Chap-man & Hall, London) (1995).

[6] A. Gelman et al. Bayesian Statistics 5 599-607, (Clarendon Press, Oxford) (1996).

Chapter8Studying the accuracy andeffectualness of closed-formwaveform models

The following published article was included in this chapter of the thesis:

• T. Damour, A. Nagar and M. Trias, Accuracy and effectualness of closed-form,f-dom waveforms for nonspinning BBHs, Physical Review D 83, 024006 (2011) [30pages]

219

Chapter9Conclusions

The coalescence of similar-mass compact binary objects are among the most importantsources of GW radiation expected to be observed with current and future interferometricdetectors. The potential astronomical GW sources within similar-mass CBCs, cover avery wide range of observable masses (from ∼ M up to ∼ 107M) and distances (fromthe galactic vicinity up to cosmological distances), which is directly related to the greatscientific impact that such observations will have on the knowledge about our Universe. Inorder to make all this possible, it is necessary to understand the dynamics that characterizesuch events and to develop data analysis techniques for the detection of signals from thenoisy strain and the extraction of reliable physical information about the emitting source.

This PhD thesis is devoted to tackle several significant data analysis (DA) problems thatneed to be solved in the current ‘pre-detection’ era. In particular, we have worked ondetection, parameter estimation and template accuracy studies following, basically, threedifferent research lines that are summarized in Table 9.1. Despite always considering thesame theoretical source [namely, the coalescence of (similar-mass) compact binary objects];the astrophysical background of the sources, their stage of evolution (and therefore, themodel waveform that can be used), the GW detector and the data analysis techniquethat have been considered are completely different in each of the cases, and cover a veryextensive range of possibilities within GW data analysis.

Source Detector Model wvf. DA technique Original contribution

SMBHs LISA PN FIMCh. 4: Impact of HHs

Ch. 5: Measuring DE EoS

gal.binaries LISA monochromatic MCMCCh. 6: MLDC search

Ch. 7: DR-MCMC algorithm

BBHs LIGO/Virgo eob,pn,phen.accuracy/

Ch. 8: Model comparisoneffectualness

Table 9.1: Summary of the source, detector, model waveform and data analysis technique used in each ofthe chapters of this thesis.

251

252 Chapter 9: Conclusions

The election of the particular projects that have been carried out during the PhD period,respond to a balance between pedagogical reasons and scientific relevance. On one hand, wehave tried to acquire a wide knowledge in GW data analysis problems by exploring differentsorts of data analysis techniques, using most of the model waveforms that currently existfor CBCs and alternating both, ground-based and space-based detectors. On the otherhand, we have tried to produce relevant scientific results within the GW community.

In particular, in Chapter 1 we give a brief introduction to the GW data analysis field.Chapter 2 is devoted to derive the expression of the GW strain measured at the detector,h(t), in terms of the original, raw output of a numerical simulation; this is, the (spherical

harmonic components of the) Regge-Wheeler and Zerilli functions, Ψ(o)`m and Ψ

(e)`m. In the

process, we are obtaining the explicit dependency of h(t) with all the orientation anglesinvolved [see Eqs. (2.52) and (2.55) for the end results].

Then, Chapter 3 introduces the basic nomenclature and main results used in GW dataanalysis, and studies the mass and distance ranges of potential GW sources within thesimilar mass CBCs, considering both ground- and space-based detectors. In particular, wedescribe a very useful way to graphically represent signal and noise amplitudes, and tointuitively visualize the expected SNR of the signals (see Figs. 3.4 and 3.5); and then weprovide astrophysical arguments to justify the expected population of (similar-mass) CBCsources. In particular,

1. LISA is expected to observe the coalescence of SMBHs from any point in our Universewith SNRs of several hundreds; these observations will help to understand the historyof galaxy mergers in our Universe and therefore the galaxy formation, but they alsocan provide very useful cosmological information;

2. LISA will observe thousands of galactic stellar-mass binaries in very early stages oftheir evolution, where their emitted GW radiation can be well defined as a simplemonochromatic signal, only modulated (in phase and amplitude) by the anisotropicsensitivity of the detector and its motion around de Sun;

3. ground-based detectors are expected to observe the late-inspiral, merger and ring-down stages of CBCs with total masses within M ∈ (1M, 500M), located in theVirgo supercluster. In this case, the expected SNRs are very low and therefore, ac-curate waveform representations will be required in order to extract the maximumSNR from the measured strain.

These first three chapters form Part I of the thesis, devoted to introductory notions. InChapters 4-8 (Part II), we present all our original scientific results, whose conclusions aresummarized in the following three sections [organized according to the three research linescarried out during the doctorate (see Tab. 9.1 for a summary)].

Parameter estimation of SMBHs inspiral signals

In the current (evaluation) stage of the LISA project, it is crucial to understand the sciencethat shall be extracted from LISA observations, thus, this is why parameter estimationstudies like the ones presented in Chapters 4 and 5 are so important.

253

We have started by studying the impact of including all the harmonics of the SMBH inspiralwaveform on the parameter estimation, instead of only using the dominant (` = 2,m = ±2)as it was done before. On one hand, the contribution of these new terms to higher harmonics(HHs) of the orbital frequency increases LISA’s mass range of detectability of SMBHs tomore massive systems, which have their dominant mode (2forb) below the low-frequencyedge of the LISA noise PSD, but not the higher harmonics (3forb, 4forb, 5forb, . . .). On theother hand, and more importantly, these new terms contribute to increase the richness ofthe waveforms, breaking some degeneracies between the parameters and therefore, signif-icantly improving the parameter estimation. The impact is found to be more importantin massive systems, M & 5× 106M, obtaining improvements of several orders of magni-tude in some parameters (like the reduced mass) and factors ∼ 10 − 50 of improvementin the most relevant parameters (from the scientific point of view), such as the luminositydistance and the sky resolution.

Moreover, we have found these improvements to make significant differences in terms ofthe fraction of events that could be identified. If we recall the results in Tab. 2 of [1] (inChapter 4), we obtain that just by the inclusion of the HHs, we go from 0.8% to 19.3% ofSMBHs binary systems with ∆Ωn < (1 × 1) for (107M + 107M) systems located atz = 1. The impact is also important when considering lower mass systems, e.g. from 10%to 35% for (106M + 105M) systems located at the same fiducial distance.

Once learnt about the importance of including HHs in the SMBH inspiral signal, we havestudied the potential scientific results that could be achieved from LISA observations ofSMBHs. We first consider different models for the massive BH merger trees [2, 3], coveringseveral possible scenarios given the limited knowledge that there exists about the actualformation history of SMBHs: small/large seeds and efficient/chaotic accretion. Results aresummarized in Tab. 5 of [4] (in Chapter 4), and our conclusions are that ∼ 30 (20) SMBHsevents should be detectable during one year of observation with LISA when consideringany small (large) seed scenario, which represents a fraction of ∼ 50% (100%) of the mergerswhich occur in the universe during Tobs. From these events, there will be about 20 sourceswith a modest distance measurement (to within 10%) and about 10 sources with a modestsky resolution (10 deg2); even more interestingly, each year LISA may observe a few sourceswith excellent sky resolution (1 deg2) and luminosity measurements (to within 1%).

In Chapter 5, we also study the possibility of using a single LISA observation of SMBH as anew tool for precision cosmography, and in particular, for measuring the dark energy (DE)equation of state, w. After establishing a criterion for localizability, we obtain (i) thefraction of events at z = 0.55, 0.7, 1 whose host galaxy could be identified; (ii) theimpact on parameter estimation of removing the degeneracies with the sky position anglesand, finally, (iii) the associated error in the estimation of w from an observation of a singleSMBH merger event. Neglecting weak lensing effects and for a fiducial redshift of z = 0.7,we find that ∆w ∼ 0.4%−5.5%, depending on the mass parameters of the event; however,weak lensing effects at these distances are expected to induce a 4% error in the estimationof DL, which translates into an 20% error in w. So, the clear conclusion here is that LISA’spotential as a new tool for precision cosmography is limited by the weak lensing effects inthe redshift range 0.55 < z < 1 rather than by the statistical errors in its measures, andtherefore, further in-depth studies are urgently needed on ways to correct for weak lensing.


Search for galactic binary systems with LISA using MCMC

Also, there has been a great effort within the LISA community [5–8] in developing searchtools. LISA will observe thousands of overlapped GW signals in a single strain time series,h(t), and it will be a data analysis task to disentangle them and extract information aboutphysical parameters. The work presented in Chapters 6 and 7 goes in this direction andin particular, we think that MCMC implementations (combined with RJMCMC in orderto deal with multiple sources in a Bayesian way; and enhanced with the DR algorithm toefficiently explore multimodal distributions) can play an important role in LISA searches.

In this thesis we have started the implementation of a MCMC-based search algorithm forsingle and multiple (but fixed number of) galactic binary signals (see Chapter 6). This isstill work in progress as we are currently testing an RJMCMC implementation that willallow us to search for an unknown number of sources and even, to search for different kindof signals simultaneously (e.g. galactic binaries, SMBH inspirals and EMRIs) and estimatethe number of them present in the data using all the power of Bayesian statistics. In parallelto this effort, we have developed a completely general and fully Markovian method toefficiently explore multimodal distributions based on the delayed rejection (see Chapter 7).This algorithm can be used in any MCMC problem presenting a target distribution with a(roughly known) multimodal structure [this was the motivation for submitting our articlepresented in Chapter 7 to a more general journal on computational statistics and dataanalysis]; and, in particular, we think that this algorithm can be applied to increase theefficiency of the MCMC-search for most of LISA sources.

Accuracy and effectualness of BBH waveform models

Ground-based detectors have reached their design sensitivity and the first direct detectionof a GW should become a fact in the next few years, with CBCs being one of the mostpromising sources. For this reason it is crucial to have available fast and accurate modelwaveforms and, in any case, to understand the validity range of any approximated modelone has at hand. The study performed in Chapter 8 tries to establish, in a formal andexhaustive way, these boundaries for the fastest, closed-form, frequency-domain modelwaveforms (PN-SPA and phenomenological models), using EOB waveforms as reference[which have shown to be more accurate than the former, though also slower to generate].

Our conclusions are that current closed-form models can be used for detection purposesin searches for CBCs with mass ratio q ∈ (1, 4) and missing less than 6% of potentialevents in case of initial detectors, and less than 9% when considering advanced detectors.The construction of new phenomenological waveform models in a wider range of massratios, could extend these results. When one is interested in extracting reliable physicalinformation [biases smaller than typical statistical errors] from the GW observations, thenmore accurate models shall be required in any case.

We also have obtained the compatibility range between EOB and PN waveforms in terms offrequency, separation distance and number of orbits before merger (see Fig. 4 in Chapter 8),finding, for 4:1 and 10:1 systems, an important gap (of several tens of orbits) between theupper limit of this range and the merger. Currently, the longest NR simulations only span∼ 10− 15 orbits.

255

We think that the results presented in this work should be useful for the LSC in order tochoose between the different waveform models that are used when performing a search.We also wish that in future works, the accuracy standards introduced in Chapter 8 and byother authors [9–12] are used rigorously (as they have been originally presented); remainingfaithful to their mathematical and physical meaning.

Future work and final remarks

All the studies performed in this PhD thesis have considered non-spinning compact objectsand, except for the galactic binary signals (which would be quasi-monochromatic anyway),we have neglected a possible equation of state of the matter conforming the compact object;which is valid in case of BHs, but just an approximation for NSs. Thus, the naturalextension of some of these results would be (i) to also consider spinning systems andfurthermore, (ii) to start investigating the potential astrophysics that can be done fromGW observations of NSs.

From the results obtained in Chapter 8, it seems clear that there is still a long roadahead in order to obtain fast and accurate template banks. On one hand, for mass ratiosgreater than 4:1, we have found a gap of several tens of orbital cycles between the high-frequency end of the PN validity range and the merger, that will have to be filled using moreaccurate analytical methods, such as EOB, given the fact that current NR simulations arelimited to ∼ 10− 15 orbits long. Also, on the other hand, the accuracy of the closed-form,frequency-domain waveforms is not good enough to exploit all the scientific potential thatground-based CBC observations have; thus, our future plans also include the investigationof building EOB-based, or NR-based, template banks.

Also, we are in the process of upgrading our MCMC-based search algorithm to a ‘DR-RJMCMC’ scheme that should represent the core of a search algorithm to deal with the“whole enchilada” of GW sources that will be an actual LISA data set, and also the nextMLDC-4 [8].

Finally, let us point out that all the studies presented in this thesis are theoretical, in thesense that no actual data from current GW detectors have been used. However, during thisperiod, the PhD candidate has also collaborated within the LSC in the search for contin-uous waves emitted by rapidly rotating NSs using the Hough transform and he also tookpart during three months of the LSC Astrowatch program for students to operate the 2 kmLIGO interferometer in Hanford (H2) between S5 and S6 science modes (February 2008 –June 2009).

During these last four years of PhD, the LSC has published 39 scientific articles in whichthe candidate appears as a co-author.


References

[1] M. Trias, A. M. Sintes, Class. Quant. Grav. 25 184032 (2008).

[2] M. Volonteri, F. Haardt, P. Madau, Astrophys. J. 582 559-573 (2003).

[3] M. C. Begelman, M. Volonteri, M. J. Rees, Mon. Not. Roy. Astron. Soc. 370 289-298 (2006).

[4] K. G. Arun et al., Class. Quant. Grav. 26 094027 (2009).

[5] K. A. Arnaud et al. [MLDC Collaboration], Class. Quant. Grav. 24 S529-S540 (2007).




[9] T. Damour, B. R. Iyer, B. S. Sathyaprakash, Phys. Rev. D 57 885-907 (1998).

[10] L. Lindblom, B. J. Owen, D. A. Brown, Phys. Rev. D 78 124020 (2008).

[11] L. Lindblom, Phys. Rev. D 80 064019 (2009).

[12] L. Lindblom, J. G. Baker, B. J. Owen, Phys. Rev. D 82 084020 (2010).

257

List of Acronyms

ADM Hamiltonian formulation of GR developed by Arnowitt, Deser and Misner.

AP asymmetrical proposal

BBH binary black hole

BBO Big Bang Observer

BH black hole

BSSN Formalism developed by T. W. Baumgarte, S. L. Shapiro, M. Shibata andT. Nakamura from 1987 to 1999, which is a modication of the ADMformalism Hamiltonian formulation of GR.

CBC compact binary coalescence

CPU central processing unit (part of a computer system)

DA data analysis

DE dark energy

DECIGO DECi-hertz Interferometer Gravitational wave Observatory

DR delayed rejection

ECP evolution of the center of the proposal

EM electromagnetic

EMRI extreme mass ratio inspiral

EOB effective-one-body

EoS equation of state

ESA European Space Agency

ET Einstein Telescope

FFT Fast Fourier Transform

FIM Fisher information matrix

FLRW Friedman-Lemaıtre-Robertson-Walker

FT Fourier transform

FWF full waveform

GR general relativity

GW gravitational wave

HH higher harmonic

INFN Istituto Nazionale di Fisica Nucleare

258 List of Acronyms

ISCO innermost stable circular orbit

LCGT Large Cryogenic Gravitational Telescope

l.h.s. left hand side

LIGO Laser Interferometer Gravitational-wave Observatory

LISA Laser Interferometer Space Antenna

LISA PE LISA Performance Evaluation (Taskforce)

LSA linearized-signal approximation

LSC LIGO Scientific Collaboration

LSO last stable orbit

LWA long wavelength approximation

MC Monte Carlo

MCMC Markov chain Monte Carlo

MH Metropolis-Hastings

ML maximum likelihood

MLDC Mock LISA Data Challenges

MM minimal match

NASA National Aeronautics and Space Administration

NR Numerical Relativity

NS neutron star

ODE ordinary differential equation

PDF probability density function

PN post-Newtonian

PSD power spectral density

r.h.s. right hand side

RJMCMC reversible jump Markov chain Monte Carlo

r.m.s. root-mean-squared

RWF restricted waveform

SMBBH supermassive binary black hole

SMBH supermassive black hole

SN supernova

259

SNR signal-to-noise ratio

SPA Stationary Phase Approximation

SSB Solar System Barycenter

TDI time delay interferometry

TOA time of arrival

TT transverse-traceless

WD white dwarf

Documents

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